<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c6">Q</font> be   <a href="qmax_1.html#NM1" title="QMAX_1:NM.1">Quantum_Mechanics</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qmax_1.html#K5" title="QMAX_1:func.5">Prop</a> <font color="Maroon" title="c6">Q</font> holds <br/> ( <font color="Olive" title="b1">p</font> <a href="qmax_1.html#R3" title="QMAX_1:pred.3">|-</a> <font color="Olive" title="b2">q</font> iff <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p3">(<span class="default"><a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font></span>)</span>,<font color="Olive" title="b1">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p3">(<span class="default"><a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font></span>)</span>,<font color="Olive" title="b2">q</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qmax_1.html#K10" title="QMAX_1:func.10">OrdRel</a> <font color="Maroon" title="c6">Q</font> )</span><br/><b>let </b><font color="Maroon" title="c7">p</font>, <font color="Maroon" title="c8">q</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qmax_1.html#K5" title="QMAX_1:func.5">Prop</a> <font color="Maroon" title="c6">Q</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">p</font> <a href="qmax_1.html#R3" title="QMAX_1:pred.3">|-</a> <font color="Maroon" title="c8">q</font> iff <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p3">(<span class="default"><a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font></span>)</span>,<font color="Maroon" title="c7">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p3">(<span class="default"><a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font></span>)</span>,<font color="Maroon" title="c8">q</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qmax_1.html#K10" title="QMAX_1:func.10">OrdRel</a> <font color="Maroon" title="c6">Q</font> )</span><br/>





<a NAME="E1:37"/>
( <font color="Maroon" title="c7">p</font> <a href="qmax_1.html#R4" title="QMAX_1:pred.4">&lt;==&gt;</a> <font color="Maroon" title="c7">p</font> &amp; <font color="Maroon" title="c8">q</font> <a href="qmax_1.html#R4" title="QMAX_1:pred.4">&lt;==&gt;</a> <font color="Maroon" title="c8">q</font> )
 
<b>by </b><i><a class="ref" href="qmax_1.html#T23" target="_self" title="QMAX_1:th.23">Th23</a></i>;<br/>
<a NAME="E2:37"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c7">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c8">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font> )
 
<b>by </b><i><a class="ref" href="qmax_1.html#D12" target="_self" title="QMAX_1:def.12">Def12</a></i>;<br/>
<a NAME="E3:37"/><b>then </b><i><font color="Green" title="E27">A1</font></i>: 
( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font></span>)</span>,<font color="Maroon" title="c7">p</font> &amp; <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font></span>)</span>,<font color="Maroon" title="c8">q</font> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/>
<a NAME="E4:37"/><i><font color="Green" title="E28">A2</font></i>: 
(  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font></span>)</span>,<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font></span>)</span> &amp;  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font></span>)</span>,<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<b>thus </b><a NAME="E5:37"/>
( <font color="Maroon" title="c7">p</font> <a href="qmax_1.html#R3" title="QMAX_1:pred.3">|-</a> <font color="Maroon" title="c8">q</font> implies <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p3">(<span class="default"><a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font></span>)</span>,<font color="Maroon" title="c7">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p3">(<span class="default"><a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font></span>)</span>,<font color="Maroon" title="c8">q</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qmax_1.html#K10" title="QMAX_1:func.10">OrdRel</a> <font color="Maroon" title="c6">Q</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p3">(<span class="default"><a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font></span>)</span>,<font color="Maroon" title="c7">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p3">(<span class="default"><a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font></span>)</span>,<font color="Maroon" title="c8">q</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qmax_1.html#K10" title="QMAX_1:func.10">OrdRel</a> <font color="Maroon" title="c6">Q</font> implies <font color="Maroon" title="c7">p</font> <a href="qmax_1.html#R3" title="QMAX_1:pred.3">|-</a> <font color="Maroon" title="c8">q</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1"/>
<font color="Maroon" title="c7">p</font> <a href="qmax_1.html#R3" title="QMAX_1:pred.3">|-</a> <font color="Maroon" title="c8">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p3">(<span class="default"><a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font></span>)</span>,<font color="Maroon" title="c7">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p3">(<span class="default"><a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font></span>)</span>,<font color="Maroon" title="c8">q</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qmax_1.html#K10" title="QMAX_1:func.10">OrdRel</a> <font color="Maroon" title="c6">Q</font></span><br/>

<a NAME="E2:37_1"/><b>then </b>
for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">q1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qmax_1.html#K5" title="QMAX_1:func.5">Prop</a> <font color="Maroon" title="c6">Q</font>  st <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font></span>)</span>,<font color="Maroon" title="c7">p</font> &amp; <font color="Olive" title="b2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font></span>)</span>,<font color="Maroon" title="c8">q</font> holds <br/><font color="Olive" title="b1">p1</font> <a href="qmax_1.html#R3" title="QMAX_1:pred.3">|-</a> <font color="Olive" title="b2">q1</font>
 
<b>by </b><i><a class="txt" href="qmax_1.html#E3:37"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="qmax_1.html#E4:37"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="qmax_1.html#T30" target="_self" title="QMAX_1:th.30">Th30</a></i>;<br/>
<b>hence </b><a NAME="E3:37_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p3">(<span class="default"><a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font></span>)</span>,<font color="Maroon" title="c7">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p3">(<span class="default"><a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font></span>)</span>,<font color="Maroon" title="c8">q</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qmax_1.html#K10" title="QMAX_1:func.10">OrdRel</a> <font color="Maroon" title="c6">Q</font>
 <b>by </b><i><a class="txt" href="qmax_1.html#E4:37"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="qmax_1.html#D13" target="_self" title="QMAX_1:def.13">Def13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E6:37"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p3">(<span class="default"><a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font></span>)</span>,<font color="Maroon" title="c7">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p3">(<span class="default"><a href="qmax_1.html#K9" title="QMAX_1:func.9">PropRel</a> <font color="Maroon" title="c6">Q</font></span>)</span>,<font color="Maroon" title="c8">q</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qmax_1.html#K10" title="QMAX_1:func.10">OrdRel</a> <font color="Maroon" title="c6">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">p</font> <a href="qmax_1.html#R3" title="QMAX_1:pred.3">|-</a> <font color="Maroon" title="c8">q</font></span><br/>

<b>hence </b><a NAME="E7:37"/>
<font color="Maroon" title="c7">p</font> <a href="qmax_1.html#R3" title="QMAX_1:pred.3">|-</a> <font color="Maroon" title="c8">q</font>
 <b>by </b><i><a class="txt" href="qmax_1.html#E3:37"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="qmax_1.html#D13" target="_self" title="QMAX_1:def.13">Def13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
