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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
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<B><FONT COLOR="#006633">Hilbert Space
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The motivation for our study of quantum logic in the <A
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<I>Hilbert space</I> (<A
HREF="http://en.wikipedia.org/wiki/Hilbert_space">Wikipedia</A>
[external],
<!--
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[external]) is a generalization of finite-dimensional
vector spaces to include vector spaces with infinite
dimensions. It provides a
foundation of quantum mechanics, and there is a strong physical and
philosophical motivation to study its properties. For example, the
properties of Hilbert space ultimately determine what kinds of
operations are theoretically possible in quantum computation.
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<TABLE BORDER=0 WIDTH="100%"><TR><TD ROWSPAN=2>
<B><FONT COLOR="#006633">Contents of this page</FONT></B>
<MENU>
<LI> <A HREF="#follow">How to Follow the Proofs</A></LI>
<LI> <A HREF="#symbol">Symbol List</A>
<!--
<FONT SIZE=-2 FACE="Comic Sans MS" COLOR=ORANGE>Revised</FONT> <FONT
SIZE=-1><I>10-Sep-2009</I></FONT>
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</LI>
<LI> <A HREF="#approaches">Two Approaches to Hilbert Space</A>
</LI>
<LI> <A HREF="#axioms">The Axioms</A>
<FONT SIZE=-2 FACE="Comic Sans MS" COLOR=ORANGE>Revised</FONT> <FONT
SIZE=-1><I>10-Sep-2009</I></FONT>
</LI>
<LI> <A HREF="#definitions">Some Definitions</A></LI>
<LI> <A HREF="#theorems">Some Theorems</A></LI>
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<LI> <A HREF="#choice">The Axiom of Choice</A></LI>
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<LI> <A HREF="#quantum">Quantum Logic</A>
<!--
<FONT SIZE=-2 FACE="Comic Sans MS" COLOR=ORANGE>Revised</FONT> <FONT
SIZE=-1><I>20-Feb-2006</I></FONT>
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</LI>
<LI> <A HREF="#next">What Next? (Orthoarguesian Law, etc.)</A></LI>
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<LI> <A HREF="#state">States on Orthomodular Lattices</A></LI>
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<P><CENTER><FONT COLOR="#006633"><I>The lyf so short,<BR>
the craft so long to lerne
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<HR NOSHADE SIZE=1><A NAME="follow"></A><B><FONT COLOR="#006633">How to
Follow the Proofs</FONT></B> We develop Hilbert space
theory as an extension of ZFC set theory, and many steps in various
proofs use results from set theory. To understand how to read the
proofs, see <A HREF="mmset.html#proofs">How Proofs Work</A> on the
Metamath Proof Explorer Home Page.
<HR NOSHADE SIZE=1><A NAME="symbol"></A><B><FONT COLOR="#006633">Symbol
List</FONT></B>
The chart below provides a quick reference for the new symbols
introduced in the Hilbert Space Explorer. The five symbols marked
"primitive" are postulated to have the properties specified by the <A
HREF="#axioms">axioms</A>, and the rest are defined in terms of them.
The complete list of all syntax elements, axioms, and definitions used
by the Hilbert Space Explorer pages, including those for the underlying
logic and ZFC set theory, is provided in the <A
HREF="mmdefinitions.html#startext">Definition List</A> (900K).
<P><CENTER><TABLE BORDER CELLSPACING=0 BGCOLOR="#EEFFFA">
<CAPTION><B>Symbol List for Hilbert Space</B></CAPTION>
<TR><TH>Symbol</TH>
<TH>Description</TH>
<TH>Link to Definition</TH></TR>
<TR><TD><IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'></TD>
<TD>Hilbert space base set</TD>
<TD>(primitive)</TD></TR>
<TR><TD><IMG SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'></TD>
<TD>vector addition</TD>
<TD>(primitive)</TD></TR>
<TR><TD><IMG SRC='_cds.gif' WIDTH=9 HEIGHT=19 ALT='.s'></TD>
<TD>scalar multiplication</TD>
<TD>(primitive)</TD></TR>
<TR><TD><IMG SRC='_0vh.gif' WIDTH=14 HEIGHT=19 ALT='0h'></TD>
<TD>zero vector</TD>
<TD>(primitive)</TD></TR>
<TR><TD><IMG SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'></TD>
<TD>inner (scalar) product</TD>
<TD>(primitive)</TD></TR>
<TR><TD><IMG SRC='_mvh.gif' WIDTH=16 HEIGHT=19 ALT='-h'></TD>
<TD>vector subtraction</TD>
<TD><A HREF="df-hvsub.html">df-hvsub</A></TD></TR>
<TR><TD><IMG SRC='_normh.gif' WIDTH=38 HEIGHT=19 ALT='normh'></TD>
<TD>norm of a vector</TD>
<TD><A HREF="df-hnorm.html">df-hnorm</A></TD></TR>
<TR><TD><IMG SRC='_cauchy.gif' WIDTH=47 HEIGHT=19 ALT='Cauchy'></TD>
<TD>set of Cauchy sequences</TD>
<TD><A HREF="df-hcau.html">df-hcau</A></TD></TR>
<TR><TD><IMG SRC='_squigv.gif' WIDTH=21 HEIGHT=19 ALT='~~>v'></TD>
<TD>convergence relation</TD>
<TD><A HREF="df-hlim.html">df-hlim</A></TD></TR>
<TR><TD><IMG SRC='_sh.gif' WIDTH=24 HEIGHT=19 ALT='SH'></TD>
<TD>set of subspaces</TD>
<TD><A HREF="df-sh.html">df-sh</A></TD></TR>
<TR><TD><IMG SRC='_scrch.gif' WIDTH=22 HEIGHT=19 ALT='CH'></TD>
<TD>set of closed subspaces</TD>
<TD><A HREF="df-ch.html">df-ch</A></TD></TR>
<TR><TD><IMG SRC='perp.gif' WIDTH=11 HEIGHT=19 ALT='_|_'></TD>
<TD>orthocomplement</TD>
<TD><A HREF="df-oc.html">df-oc</A></TD></TR>
<TR><TD><IMG SRC='_plh.gif' WIDTH=24 HEIGHT=19 ALT='+H'></TD>
<TD>subspace sum</TD>
<TD><A HREF="df-shs.html">df-shs</A></TD></TR>
<TR><TD><IMG SRC='_span.gif' WIDTH=31 HEIGHT=19 ALT='span'></TD>
<TD>subspace span</TD>
<TD><A HREF="df-span.html">df-span</A></TD></TR>
<TR><TD><IMG SRC='_veeh.gif' WIDTH=21 HEIGHT=19 ALT='vH'></TD>
<TD>join</TD>
<TD><A HREF="df-chj.html">df-chj</A></TD></TR>
<TR><TD><IMG SRC='_bigveeh.gif' WIDTH=23 HEIGHT=19 ALT='\/H'></TD>
<TD>supremum</TD>
<TD><A HREF="df-chsup.html">df-chsup</A></TD></TR>
<TR><TD><IMG SRC='_0h.gif' WIDTH=20 HEIGHT=19 ALT='0H'></TD>
<TD>zero subspace</TD>
<TD><A HREF="df-ch0.html">df-ch0</A></TD></TR>
<TR><TD><IMG SRC='_cch.gif' WIDTH=23 HEIGHT=19 ALT='C_H'></TD>
<TD>commutes relation</TD>
<TD><A HREF="df-cm.html">df-cm</A></TD></TR>
<TR><TD><IMG SRC='_plop.gif' WIDTH=25 HEIGHT=19 ALT='+op'></TD>
<TD>operator sum;<BR>definition of "operator"</TD>
<TD><A HREF="df-hosum.html">df-hosum</A></TD></TR>
<TR><TD><IMG SRC='_cdop.gif' WIDTH=16 HEIGHT=19 ALT='.op'></TD>
<TD>operator scalar product</TD>
<TD><A HREF="df-homul.html">df-homul</A></TD></TR>
<TR><TD><IMG SRC='_mop.gif' WIDTH=23 HEIGHT=19 ALT='-op'></TD>
<TD>operator difference</TD>
<TD><A HREF="df-hodif.html">df-hodif</A></TD></TR>
<TR><TD><IMG SRC='_plfn.gif' WIDTH=24 HEIGHT=19 ALT='+fn'></TD>
<TD>functional sum;<BR>definition of "functional"</TD>
<TD><A HREF="df-hfsum.html">df-hfsum</A></TD></TR>
<TR><TD><IMG SRC='_cdfn.gif' WIDTH=15 HEIGHT=19 ALT='.fn'></TD>
<TD>functional scalar product</TD>
<TD><A HREF="df-hfmul.html">df-hfmul</A></TD></TR>
<!--
<TR><TD><IMG SRC='_0op.gif' WIDTH=21 HEIGHT=19 ALT='0op'></TD>
-->
<TR><TD>0<SUB>hop</SUB> </TD>
<TD>zero operator</TD>
<TD><A HREF="df-h0op.html">df-h0op</A></TD></TR>
<TR><TD><IMG SRC='_iop.gif' WIDTH=18 HEIGHT=19 ALT='iop'></TD>
<TD>identity operator</TD>
<TD><A HREF="df-iop.html">df-iop</A></TD></TR>
<TR><TD>proj</TD>
<TD>projection operator (projector)</TD>
<TD><A HREF="df-pj.html">df-pj</A></TD></TR>
<TR><TD>norm<SUB>op</SUB></TD>
<TD>norm of an operator</TD>
<TD><A HREF="df-nmop.html">df-nmop</A></TD></TR>
<TR><TD>ConOp</TD>
<TD>set of continuous operators</TD>
<TD><A HREF="df-cnop.html">df-cnop</A></TD></TR>
<TR><TD>LinOp</TD>
<TD>set of linear operators</TD>
<TD><A HREF="df-lnop.html">df-lnop</A></TD></TR>
<TR><TD>BndLinOp</TD>
<TD>set of bounded linear operators</TD>
<TD><A HREF="df-bdop.html">df-bdop</A></TD></TR>
<TR><TD>UniOp</TD>
<TD>set of unitary operators</TD>
<TD><A HREF="df-unop.html">df-unop</A></TD></TR>
<TR><TD>HrmOp</TD>
<TD>set of Hermitian operators</TD>
<TD><A HREF="df-hmop.html">df-hmop</A></TD></TR>
<TR><TD>norm<SUB>fn</SUB></TD>
<TD>norm of a functional</TD>
<TD><A HREF="df-nmfn.html">df-nmfn</A></TD></TR>
<TR><TD>null</TD>
<TD>null space of a functional</TD>
<TD><A HREF="df-nlfn.html">df-nlfn</A></TD></TR>
<TR><TD>ConFn</TD>
<TD>set of continuous functionals</TD>
<TD><A HREF="df-cnfn.html">df-cnfn</A></TD></TR>
<TR><TD>LinFn</TD>
<TD>set of linear functionals</TD>
<TD><A HREF="df-lnfn.html">df-lnfn</A></TD></TR>
<TR><TD>adj<SUB>h</SUB></TD>
<TD>adjoint of an operator</TD>
<TD><A HREF="df-adjh.html">df-adjh</A></TD></TR>
<TR><TD>bra</TD>
<TD>Dirac "bra" of a vector</TD>
<TD><A HREF="df-bra.html">df-bra</A></TD></TR>
<TR><TD><IMG SRC='_leop.gif' WIDTH=24 HEIGHT=19 ALT='<_op'></TD>
<TD>ordering relation for positive operators</TD>
<TD><A HREF="df-leop.html">df-leop</A></TD></TR>
<TR><TD>ketbra</TD>
<TD>Dirac "ket-bra" (outer product) of two vectors</TD>
<TD><A HREF="df-kb.html">df-kb</A></TD></TR>
<TR><TD>eigvec</TD>
<TD>eigenvectors of an operator</TD>
<TD><A HREF="df-eigvec.html">df-eigvec</A></TD></TR>
<TR><TD>eigval</TD>
<TD>eigenvalue of an eigenvector</TD>
<TD><A HREF="df-eigval.html">df-eigval</A></TD></TR>
<TR><TD><IMG SRC='clambda.gif' WIDTH=11 HEIGHT=19 ALT='Lambda'></TD>
<TD>spectrum of an operator</TD>
<TD><A HREF="df-spec.html">df-spec</A></TD></TR>
<TR><TD><IMG SRC='_states.gif' WIDTH=40 HEIGHT=19 ALT='States'></TD>
<TD>set of states</TD>
<TD><A HREF="df-st.html">df-st</A></TD></TR>
<TR><TD>CHStates</TD>
<TD>set of (Mayet's) Hilbert-space-valued states</TD>
<TD><A HREF="df-hst.html">df-hst</A></TD></TR>
<TR><TD><IMG SRC='_atoms.gif' WIDTH=40 HEIGHT=19 ALT='Atoms'></TD>
<TD>set of atoms</TD>
<TD><A HREF="df-at.html">df-at</A></TD></TR>
<TR><TD><IMG SRC='lessdot.gif' WIDTH=11 HEIGHT=19 ALT='<o'></TD>
<TD>covering relation</TD>
<TD><A HREF="df-cv.html">df-cv</A></TD></TR>
<TR><TD><IMG SRC='_mh.gif' WIDTH=27 HEIGHT=19 ALT='MH'></TD>
<TD>modular pair relation</TD>
<TD><A HREF="df-md.html">df-md</A></TD></TR>
<TR><TD><IMG SRC='_mhast.gif' WIDTH=27 HEIGHT=19 ALT='MH*'></TD>
<TD>dual modular pair relation</TD>
<TD><A HREF="df-dmd.html">df-dmd</A></TD></TR>
</TABLE></CENTER>
<P>
<P><HR NOSHADE SIZE=1><A NAME="approaches"></A><B><FONT
COLOR="#006633">Two Approaches to Hilbert
Space</FONT></B> There are several ways to develop the
theory of Hilbert spaces. The purest way, philosophically, is to define
the class of all Hilbert spaces and use only the axioms of ZFC set
theory to derive its properties. That way we need to assume only the
axioms of ZFC (which in principle is all that is needed for essentially
all of mathematics, including the theory of Hilbert spaces). This is
done in the Metamath Proof Explorer with definition <A
HREF="df-hl.html">df-hl</A>.
<P>However, we chose separate axioms for the Hilbert Space Explorer for
several reasons. A practical problem with the pure ZFC approach is that
theorems becomes somewhat awkward to state and prove, since they usually
need additional hypotheses. Compare, for example, the ZFC-derived <A
HREF="hlcom.html">hlcom</A> with the Hilbert Space Explorer axiom <A
HREF="ax-hvcom.html">ax-hvcom</A>. Another advantage for a newcomer is that
the Hilbert Space Explorer states outright all of its axioms, so there
is nothing else to learn (aside from standard set theory tools to
manipulate them). In the Metamath Proof Explorer, on the other hand,
one needs to become familiar with the hierarchy of groups, topologies,
vector spaces, metric spaces, normed vector spaces, and Banach spaces
that leads to Hilbert spaces.
<P> If we want to use the Hilbert Space Explorer with any <I>fixed</I>
Hilbert space, such as the set of complex numbers (which, as it turns
out, is an example of a Hilbert space - see theorem <A
HREF="cnhl.html">cnhl</A>), a simple change to the axiomatization will
convert all theorems in the Hilbert Space Explorer to pure ZFC theorems.
A description of how this can be done is given in the comment for axiom
<A HREF="ax-hilex.html">ax-hilex</A>. On the other hand, if we want to prove
theorems involving relationships between Hilbert spaces, the Hilbert
Space Explorer may not be not suitable, but rewriting its proofs for the
general ZFC approach as needed is relatively straightforward.
(Actually, many such theorems can still be done in the Hilbert Space
Explorer itself using subspaces, each of which acts like a stand-alone
Hilbert Space.)
<P><HR NOSHADE SIZE=1><A NAME="axioms"></A><B><FONT COLOR="#006633">The
Axioms</FONT></B> In our separately axiomatized
approach of the Hilbert Space Explorer,
we postulate the existence of a new primitive fixed object, <IMG
SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'> (<A
HREF="chil.html">chil</A>), called the
Hilbert space base set, and add to ZFC set
theory explicit axioms for the properties of <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'>.
The members of
<IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'>
are called vectors, and they have the same
properties as the vectors you normally find in any linear algebra
textbook, except that the dimension (the number of basis vectors) is not
specified and may be infinite. In addition to
<IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'>,
we postulate
the existence and properties of 4 more objects: a fixed zero vector
<IMG SRC='_0vh.gif' WIDTH=14 HEIGHT=19 ALT='0h'>
(<A HREF="c0v.html">c0v</A>); the operations of vector
addition
<IMG SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'>
(<A HREF="cva.html">cva</A>) and scalar
multiplication
<IMG SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'>
(<A HREF="csm.html">csm</A>); and finally, an inner product operation
<IMG SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'>
(<A HREF="csp.html">csp</A>). The five objects
<IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'>,
<IMG SRC='_0vh.gif' WIDTH=14 HEIGHT=19 ALT='0h'>,
<IMG SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'>,
<IMG SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'>,
and
<IMG SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'>
are the complete
set of objects needed to describe Hilbert space. We will encounter
other objects as well, but all of them are defined either in terms of these
five, or as specific sets of set theory. For example, the object
<IMG SRC='bbc.gif' WIDTH=12 HEIGHT=19 ALT='CC'>
(the set of complex numbers <A HREF="cc.html">cc</A>) is
defined as a specific set of set theory.
<P>The page for each axiom below is accompanied by a precise breakdown
of its syntax. You can break the
syntax down into as much detail as you want by using the hyperlinks
in the syntax breakdown chart. Note our use of the
notation "<IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'>
<IMG SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'>
<IMG SRC='_cb.gif'
WIDTH=12 HEIGHT=19 ALT='B'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'>" instead of the more common notation "<IMG
SRC='langle.gif' WIDTH=4 HEIGHT=19 ALT='<.' ><IMG
SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'><IMG SRC='comma.gif'
WIDTH=4 HEIGHT=19 ALT=','><IMG SRC='_cb.gif' WIDTH=12
HEIGHT=19 ALT='B'><IMG SRC='rangle.gif' WIDTH=4 HEIGHT=19
ALT='>.' >" for inner products; the latter would
conflict with our notation for ordered pairs <A HREF="cop.html">cop</A>.
<P><CENTER><TABLE BORDER CELLSPACING=0 BGCOLOR="#EEFFFA">
<CAPTION><B>Axioms for Hilbert Space</B></CAPTION>
<TR ALIGN=LEFT><TD><A HREF="ax-hilex.html">ax-hilex</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG
SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='in.gif'
WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='cv.gif' WIDTH=12 HEIGHT=19
ALT='V'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hfvadd.html">ax-hfvadd</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG
SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'> <IMG SRC='colon.gif'
WIDTH=4 HEIGHT=19 ALT=':'><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'> <IMG
SRC='times.gif' WIDTH=9 HEIGHT=19 ALT='X.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'><IMG SRC='longrightarrow.gif' WIDTH=23 HEIGHT=19 ALT='-->'
ALIGN=TOP><IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hvcom.html">ax-hvcom</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19
ALT='/\'> <IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'>
<IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='to.gif' WIDTH=15 HEIGHT=19 ALT='->'>
<IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='_ca.gif'
WIDTH=11 HEIGHT=19 ALT='A'> <IMG SRC='_pvh.gif' WIDTH=18 HEIGHT=19
ALT='+h'> <IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'> <IMG
SRC='eq.gif' WIDTH=12 HEIGHT=19 ALT='='> <IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'
ALIGN=TOP> <IMG SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'> <IMG
SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'
ALIGN=TOP></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hvass.html">ax-hvass</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19
ALT='/\'> <IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'>
<IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19
ALT='/\'> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'>
<IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='to.gif' WIDTH=15 HEIGHT=19 ALT='->'>
<IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'
ALIGN=TOP> <IMG SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'> <IMG
SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'> <IMG SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'
ALIGN=TOP> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'><IMG
SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'> <IMG SRC='eq.gif' WIDTH=12
HEIGHT=19 ALT='='> <IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'> <IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'
ALIGN=TOP> <IMG SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'> <IMG
SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hv0cl.html">ax-hv0cl</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG
SRC='_0vh.gif' WIDTH=14 HEIGHT=19 ALT='0h'> <IMG SRC='in.gif'
WIDTH=10 HEIGHT=19
ALT='e.'> <IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'
ALIGN=TOP></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hvaddid.html">ax-hvaddid</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19
ALT='A'> <IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'>
<IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='to.gif'
WIDTH=15 HEIGHT=19 ALT='->'> <IMG SRC='lp.gif' WIDTH=5 HEIGHT=19
ALT='('><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'> <IMG SRC='_0vh.gif'
WIDTH=14 HEIGHT=19 ALT='0h'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='eq.gif' WIDTH=12 HEIGHT=19 ALT='='> <IMG
SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hfvmul.html">ax-hfvmul</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG
SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'> <IMG SRC='colon.gif'
WIDTH=4 HEIGHT=19 ALT=':'><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='bbc.gif' WIDTH=12 HEIGHT=19 ALT='CC'> <IMG
SRC='times.gif' WIDTH=9 HEIGHT=19 ALT='X.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'><IMG SRC='longrightarrow.gif' WIDTH=23 HEIGHT=19 ALT='-->'
ALIGN=TOP><IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hvmulid.html">ax-hvmulid</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19
ALT='A'> <IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'>
<IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='to.gif'
WIDTH=15 HEIGHT=19 ALT='->'> <IMG SRC='lp.gif' WIDTH=5 HEIGHT=19
ALT='('><IMG SRC='1.gif' WIDTH=7 HEIGHT=19 ALT='1'> <IMG
SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'> <IMG SRC='_ca.gif'
WIDTH=11 HEIGHT=19 ALT='A'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='eq.gif' WIDTH=12 HEIGHT=19 ALT='='> <IMG
SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hvmulass.html">ax-hvmulass</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='bbc.gif' WIDTH=12
HEIGHT=19 ALT='CC'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19 ALT='/\'
ALIGN=TOP> <IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='bbc.gif' WIDTH=12
HEIGHT=19 ALT='CC'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19 ALT='/\'
ALIGN=TOP> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='to.gif' WIDTH=15 HEIGHT=19 ALT='->'>
<IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'
ALIGN=TOP> <IMG SRC='cdot.gif' WIDTH=4 HEIGHT=19 ALT='x.'> <IMG
SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'> <IMG SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'
ALIGN=TOP> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'><IMG
SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'> <IMG SRC='eq.gif' WIDTH=12
HEIGHT=19 ALT='='> <IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'> <IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'
ALIGN=TOP> <IMG SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'> <IMG
SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hvdistr1.html">ax-hvdistr1</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='bbc.gif' WIDTH=12
HEIGHT=19 ALT='CC'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19 ALT='/\'
ALIGN=TOP> <IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19
ALT='/\'> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'>
<IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='to.gif' WIDTH=15 HEIGHT=19 ALT='->'>
<IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='_ca.gif'
WIDTH=11 HEIGHT=19 ALT='A'> <IMG SRC='_cdh.gif' WIDTH=9 HEIGHT=19
ALT='.h'> <IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG
SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'> <IMG SRC='_pvh.gif' WIDTH=18
HEIGHT=19 ALT='+h'> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19
ALT='C'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'><IMG
SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'> <IMG SRC='eq.gif' WIDTH=12
HEIGHT=19 ALT='='> <IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG
SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG SRC='_cdh.gif' WIDTH=9
HEIGHT=19 ALT='.h'> <IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'> <IMG
SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'> <IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'
ALIGN=TOP> <IMG SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'> <IMG
SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hvdistr2.html">ax-hvdistr2</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
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SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='bbc.gif' WIDTH=12
HEIGHT=19 ALT='CC'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19 ALT='/\'
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SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='bbc.gif' WIDTH=12
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SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='to.gif' WIDTH=15 HEIGHT=19 ALT='->'>
<IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'
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SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'><IMG SRC='rp.gif' WIDTH=5
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SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'> <IMG SRC='eq.gif' WIDTH=12
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HEIGHT=19 ALT='.h'> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'> <IMG
SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'> <IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'
ALIGN=TOP> <IMG SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'> <IMG
SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hvmul0.html">ax-hvmul0</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19
ALT='A'> <IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'>
<IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='to.gif'
WIDTH=15 HEIGHT=19 ALT='->'> <IMG SRC='lp.gif' WIDTH=5 HEIGHT=19
ALT='('><IMG SRC='0.gif' WIDTH=8 HEIGHT=19 ALT='0'> <IMG
SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'> <IMG SRC='_ca.gif'
WIDTH=11 HEIGHT=19 ALT='A'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='eq.gif' WIDTH=12 HEIGHT=19 ALT='='> <IMG
SRC='_0vh.gif' WIDTH=14 HEIGHT=19 ALT='0h'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hfi.html">ax-hfi</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG
SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'> <IMG SRC='colon.gif'
WIDTH=4 HEIGHT=19 ALT=':'><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'> <IMG
SRC='times.gif' WIDTH=9 HEIGHT=19 ALT='X.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'><IMG SRC='longrightarrow.gif' WIDTH=23 HEIGHT=19 ALT='-->'
ALIGN=TOP><IMG SRC='bbc.gif' WIDTH=12 HEIGHT=19 ALT='CC'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-his1.html">ax-his1</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19
ALT='/\'> <IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'>
<IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='to.gif' WIDTH=15 HEIGHT=19 ALT='->'>
<IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='_ca.gif'
WIDTH=11 HEIGHT=19 ALT='A'> <IMG SRC='_cdih.gif' WIDTH=13 HEIGHT=19
ALT='.ih'> <IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'> <IMG
SRC='eq.gif' WIDTH=12 HEIGHT=19 ALT='='> <IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='ast.gif' WIDTH=7 HEIGHT=19 ALT='*'
ALIGN=TOP><IMG SRC='backtick.gif' WIDTH=4 HEIGHT=19 ALT='`'><IMG
SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='_cb.gif' WIDTH=12
HEIGHT=19 ALT='B'> <IMG SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'
ALIGN=TOP> <IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'><IMG
SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'
ALIGN=TOP></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-his2.html">ax-his2</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19
ALT='/\'> <IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'>
<IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19
ALT='/\'> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'>
<IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='to.gif' WIDTH=15 HEIGHT=19 ALT='->'>
<IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'
ALIGN=TOP> <IMG SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'> <IMG
SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'> <IMG SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'
ALIGN=TOP> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'><IMG
SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'> <IMG SRC='eq.gif' WIDTH=12
HEIGHT=19 ALT='='> <IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG
SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG SRC='_cdih.gif'
WIDTH=13 HEIGHT=19 ALT='.ih'> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19
ALT='C'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'> <IMG
SRC='plus.gif' WIDTH=13 HEIGHT=19 ALT='+'> <IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'
ALIGN=TOP> <IMG SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'> <IMG
SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-his3.html">ax-his3</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='bbc.gif' WIDTH=12
HEIGHT=19 ALT='CC'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19 ALT='/\'
ALIGN=TOP> <IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19
ALT='/\'> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'>
<IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='to.gif' WIDTH=15 HEIGHT=19 ALT='->'>
<IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'
ALIGN=TOP> <IMG SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'> <IMG
SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'> <IMG SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'
ALIGN=TOP> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'><IMG
SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'> <IMG SRC='eq.gif' WIDTH=12
HEIGHT=19 ALT='='> <IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='cdot.gif' WIDTH=4 HEIGHT=19 ALT='x.'> <IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'
ALIGN=TOP> <IMG SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'> <IMG
SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-his4.html">ax-his4</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19
ALT='/\'> <IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'>
<IMG SRC='ne.gif' WIDTH=12 HEIGHT=19 ALT='=/='> <IMG SRC='_0vh.gif'
WIDTH=14 HEIGHT=19 ALT='0h'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='to.gif' WIDTH=15 HEIGHT=19 ALT='->'>
<IMG SRC='0.gif' WIDTH=8 HEIGHT=19 ALT='0'> <IMG SRC='lt.gif'
WIDTH=11 HEIGHT=19 ALT='<'> <IMG SRC='lp.gif' WIDTH=5 HEIGHT=19
ALT='('><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'> <IMG SRC='_ca.gif'
WIDTH=11 HEIGHT=19 ALT='A'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'
ALIGN=TOP></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hcompl.html">ax-hcompl</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='_cf.gif' WIDTH=13 HEIGHT=19
ALT='F'> <IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'>
<IMG SRC='_cauchy.gif' WIDTH=47 HEIGHT=19 ALT='Cauchy'> <IMG
SRC='to.gif' WIDTH=15 HEIGHT=19 ALT='->'> <IMG SRC='exists.gif'
WIDTH=9 HEIGHT=19 ALT='E.'><IMG SRC='_x.gif' WIDTH=10 HEIGHT=19
ALT='x'> <IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'>
<IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='_cf.gif'
WIDTH=13 HEIGHT=19 ALT='F'> <IMG SRC='_squigv.gif' WIDTH=21 HEIGHT=19
ALT='~~>v'> <IMG SRC='_x.gif' WIDTH=10 HEIGHT=19 ALT='x'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'></TD></TR>
</TABLE></CENTER>
<P><I>Comments on the axioms.</I> The first axiom just says that the
primitive class
<IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'>
exists (is a member of the universe of sets <I>V</I>).
The next 11 axioms are the axioms for any vector space with
an unspecified dimension; they are the same as those you would find in
any linear algebra book, except for the notation and possibly their
precise form.
<P>The next 5 axioms show the properties of the special inner product
<IMG SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'>. The official
name for this inner product is a "sesquilinear Hermitian
mapping". (Sesquilinear means "one-and-a-half linear," i.e.,
antilinear in the first argument and linear in the second.) The symbol
<IMG SRC='ast.gif' WIDTH=7 HEIGHT=19 ALT='*'> in Axiom <A
HREF="ax-his1.html">ax-his1</A> is the complex conjugate (<A
HREF="cjval.html">cjval</A>). See <A
HREF="mmset.html#function">Notation for Function Values</A> for an
explanation of why we use this notation rather than the standard
superscript asterisk used in textbooks; this will help you understand
some of our other non-standard notation as well.
<P>The last axiom, which is the most important and also the most
complicated, is called the Completeness Axiom, and is shown above using
abbreviations. You can click on its links to expand the abbreviations.
It basically says that the limit of any converging ("Cauchy")
sequence of vectors in Hilbert space converges to a vector in Hilbert
space. To understand what completeness means, consider this analogy:
the sequence 3, 3.1, 3.14, 3.1415, 3.14159... converges to pi. This is
a converging sequence of rational numbers, but it converges to something
that is not a rational number, meaning the set of rational numbers is
<I>not</I> complete. The set of real numbers, on the other hand,
<I>is</I> complete, because all converging sequences of real numbers
converge to a real number.
<P><HR NOSHADE SIZE=1><A NAME="definitions"></A><B><FONT
COLOR="#006633">Some Definitions</FONT></B> Here we
show explicitly a few of the definitions you will encounter in our
Hilbert space proofs. The complete list is given at the top of this
page. We typically define new symbols as a self-contained objects, which
can make their definitions seem unnecessarily complicated, but usually
their Descriptions point to simpler theorems showing their values or
other properties. For example, the vector subtraction operation <IMG
SRC='_mvh.gif' WIDTH=16 HEIGHT=19 ALT='-h'> is formally a set
of ordered pairs as shown below, but its value is just <IMG
SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG SRC='_pvh.gif'
WIDTH=18 HEIGHT=19 ALT='+h'> <IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='shortminus.gif' WIDTH=8 HEIGHT=19
ALT='-u'><IMG SRC='1.gif' WIDTH=7 HEIGHT=19 ALT='1'>
<IMG SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'> <IMG
SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'><IMG SRC='rp.gif'
WIDTH=5 HEIGHT=19 ALT=')'> as can be seen from theorem <A
HREF="hvsubval.html">hvsubval</A>.
<P><CENTER><TABLE BORDER CELLSPACING=0 BGCOLOR="#EEFFFA">
<CAPTION><B>Some Definitions for Hilbert Space</B></CAPTION>
<TR ALIGN=LEFT><TD>The vector subtraction operation
<A HREF="df-hvsub.html">df-hvsub</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG
SRC='_mvh.gif' WIDTH=16 HEIGHT=19 ALT='-h'> <IMG SRC='eq.gif'
WIDTH=12 HEIGHT=19 ALT='='> <IMG SRC='lbrace.gif' WIDTH=6 HEIGHT=19
ALT='{'><IMG SRC='langle.gif' WIDTH=4 HEIGHT=19 ALT='<.'
ALIGN=TOP><IMG SRC='langle.gif' WIDTH=4 HEIGHT=19 ALT='<.'><IMG
SRC='_x.gif' WIDTH=10 HEIGHT=19 ALT='x'><IMG SRC='comma.gif' WIDTH=4
HEIGHT=19 ALT=','><IMG SRC='_y.gif' WIDTH=9 HEIGHT=19 ALT='y'
ALIGN=TOP><IMG SRC='rangle.gif' WIDTH=4 HEIGHT=19 ALT='>.'><IMG
SRC='comma.gif' WIDTH=4 HEIGHT=19 ALT=','><IMG SRC='_z.gif' WIDTH=9
HEIGHT=19 ALT='z'><IMG SRC='rangle.gif' WIDTH=4 HEIGHT=19 ALT='>.'
ALIGN=TOP> <IMG SRC='vert.gif' WIDTH=3 HEIGHT=19 ALT='|'> <IMG
SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_x.gif' WIDTH=10 HEIGHT=19 ALT='x'
ALIGN=TOP> <IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG
SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='wedge.gif'
WIDTH=11 HEIGHT=19 ALT='/\'> <IMG SRC='_y.gif' WIDTH=9 HEIGHT=19
ALT='y'> <IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'>
<IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif'
WIDTH=5 HEIGHT=19 ALT=')'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19
ALT='/\'> <IMG SRC='_z.gif' WIDTH=9 HEIGHT=19 ALT='z'> <IMG
SRC='eq.gif' WIDTH=12 HEIGHT=19 ALT='='> <IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_x.gif' WIDTH=10 HEIGHT=19 ALT='x'
ALIGN=TOP> <IMG SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'> <IMG
SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='shortminus.gif'
WIDTH=8 HEIGHT=19 ALT='-u'><IMG SRC='1.gif' WIDTH=7 HEIGHT=19 ALT='1'
ALIGN=TOP> <IMG SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'> <IMG
SRC='_y.gif' WIDTH=9 HEIGHT=19 ALT='y'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'><IMG
SRC='rbrace.gif' WIDTH=6 HEIGHT=19 ALT='}'></TD></TR>
<TR ALIGN=LEFT><TD>The norm of a vector
<A HREF="df-hnorm.html">df-hnorm</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG
SRC='_normh.gif' WIDTH=38 HEIGHT=19 ALT='normh'> <IMG SRC='eq.gif'
WIDTH=12 HEIGHT=19 ALT='='> <IMG SRC='lbrace.gif' WIDTH=6 HEIGHT=19
ALT='{'><IMG SRC='langle.gif' WIDTH=4 HEIGHT=19 ALT='<.'
ALIGN=TOP><IMG SRC='_x.gif' WIDTH=10 HEIGHT=19 ALT='x'><IMG
SRC='comma.gif' WIDTH=4 HEIGHT=19 ALT=','><IMG SRC='_y.gif' WIDTH=9
HEIGHT=19 ALT='y'><IMG SRC='rangle.gif' WIDTH=4 HEIGHT=19 ALT='>.'
ALIGN=TOP> <IMG SRC='vert.gif' WIDTH=3 HEIGHT=19 ALT='|'> <IMG
SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='_x.gif' WIDTH=10
HEIGHT=19 ALT='x'> <IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'
ALIGN=TOP> <IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'> <IMG
SRC='wedge.gif' WIDTH=11 HEIGHT=19 ALT='/\'> <IMG SRC='_y.gif'
WIDTH=9 HEIGHT=19 ALT='y'> <IMG SRC='eq.gif' WIDTH=12 HEIGHT=19
ALT='='> <IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG
SRC='surd.gif' WIDTH=14 HEIGHT=19 ALT='sqrt'><IMG SRC='backtick.gif'
WIDTH=4 HEIGHT=19 ALT='`'><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_x.gif' WIDTH=10 HEIGHT=19 ALT='x'> <IMG
SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'> <IMG SRC='_x.gif'
WIDTH=10 HEIGHT=19 ALT='x'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'><IMG
SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'><IMG SRC='rbrace.gif' WIDTH=6
HEIGHT=19 ALT='}'></TD></TR>
<TR ALIGN=LEFT><TD>The set of all Cauchy sequences
<A HREF="df-hcau.html">df-hcau</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG
SRC='_cauchy.gif' WIDTH=47 HEIGHT=19 ALT='Cauchy'> <IMG SRC='eq.gif'
WIDTH=12 HEIGHT=19 ALT='='> <IMG SRC='lbrace.gif' WIDTH=6 HEIGHT=19
ALT='{'><IMG SRC='_f.gif' WIDTH=9 HEIGHT=19 ALT='f'> <IMG
SRC='vert.gif' WIDTH=3 HEIGHT=19 ALT='|'> <IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_f.gif' WIDTH=9 HEIGHT=19 ALT='f'
ALIGN=TOP><IMG SRC='colon.gif' WIDTH=4 HEIGHT=19 ALT=':'><IMG
SRC='bbn.gif' WIDTH=12 HEIGHT=19 ALT='NN'><IMG
SRC='longrightarrow.gif' WIDTH=23 HEIGHT=19 ALT='-->'><IMG
SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='wedge.gif'
WIDTH=11 HEIGHT=19 ALT='/\'> <IMG SRC='forall.gif' WIDTH=10 HEIGHT=19
ALT='A.'><IMG SRC='_x.gif' WIDTH=10 HEIGHT=19 ALT='x'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='bbr.gif' WIDTH=13
HEIGHT=19 ALT='RR'><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='0.gif' WIDTH=8 HEIGHT=19 ALT='0'> <IMG
SRC='lt.gif' WIDTH=11 HEIGHT=19 ALT='<'> <IMG SRC='_x.gif'
WIDTH=10 HEIGHT=19 ALT='x'> <IMG SRC='to.gif' WIDTH=15 HEIGHT=19
ALT='->'> <IMG SRC='exists.gif' WIDTH=9 HEIGHT=19 ALT='E.'
ALIGN=TOP><IMG SRC='_y.gif' WIDTH=9 HEIGHT=19 ALT='y'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='bbn.gif' WIDTH=12
HEIGHT=19 ALT='NN'><IMG SRC='forall.gif' WIDTH=10 HEIGHT=19 ALT='A.'
ALIGN=TOP><IMG SRC='_z.gif' WIDTH=9 HEIGHT=19 ALT='z'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='bbn.gif' WIDTH=12