<?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>;<br/>

<a NAME="E1:41"/><i><font color="Green" title="E29">A1</font></i>: 
 <a href="qmax_1.html#K10" title="QMAX_1:func.10">OrdRel</a> <font color="Maroon" title="c6">Q</font> <a href="orders_1.html#R2" title="ORDERS_1:pred.2">partially_orders</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:41_1"/>
 <a href="qmax_1.html#K10" title="QMAX_1:func.10">OrdRel</a> <font color="Maroon" title="c6">Q</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_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>
  <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D7" title="ORDERS_1:def.7">ORDERS_1:def 7</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a><br/>

<b>assume </b><a NAME="E1:41_1_1"/>
<font color="Maroon" title="c7">x</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>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">p</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><b> such that </b><br/><a NAME="E3:41_1_1"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">p</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T45" title="EQREL_1:th.45">EQREL_1:45</a></i>;<br/>
<a NAME="E4:41_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="qmax_1.html#R3" title="QMAX_1:pred.3">|-</a> <font color="Maroon" title="c8">p</font>
 
<b>by </b><i><a class="ref" href="qmax_1.html#T21" target="_self" title="QMAX_1:th.21">Th21</a></i>;<br/>
<b>hence </b><a NAME="E5:41_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c7">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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#E3:41_1_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="qmax_1.html#T32" target="_self" title="QMAX_1:th.32">Th32</a></i>;<br/>


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

<b>thus </b><a NAME="E2:41_1"/>
 <a href="qmax_1.html#K10" title="QMAX_1:func.10">OrdRel</a> <font color="Maroon" title="c6">Q</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a><br/><b>let </b><font color="Maroon" title="c8">y</font>, <font color="Maroon" title="c9">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:41_1_2"/><i><font color="Green" title="E30">A3</font></i>: 
( <font color="Maroon" title="c7">x</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; <font color="Maroon" title="c8">y</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; <font color="Maroon" title="c9">z</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>and </b><br/><a NAME="E2:41_1_2"/><i><font color="Green" title="E31">A4</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c10">p</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><b> such that </b><br/><a NAME="E4:41_1_2"/><i><font color="Green" title="E32">A5</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c10">p</font>
 <b>by </b><i><a class="txt" href="qmax_1.html#E1:41_1_2"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="eqrel_1.html#T45" title="EQREL_1:th.45">EQREL_1:45</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">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><b> such that </b><br/><a NAME="E6:41_1_2"/><i><font color="Green" title="E33">A6</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c11">q</font>
 <b>by </b><i><a class="txt" href="qmax_1.html#E1:41_1_2"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="eqrel_1.html#T45" title="EQREL_1:th.45">EQREL_1:45</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">r</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><b> such that </b><br/><a NAME="E8:41_1_2"/><i><font color="Green" title="E34">A7</font></i>: 
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c12">r</font>
 <b>by </b><i><a class="txt" href="qmax_1.html#E1:41_1_2"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="eqrel_1.html#T45" title="EQREL_1:th.45">EQREL_1:45</a></i>;<br/>
<a NAME="E9:41_1_2"/>
( <font color="Maroon" title="c10">p</font> <a href="qmax_1.html#R3" title="QMAX_1:pred.3">|-</a> <font color="Maroon" title="c11">q</font> &amp; <font color="Maroon" title="c11">q</font> <a href="qmax_1.html#R3" title="QMAX_1:pred.3">|-</a> <font color="Maroon" title="c12">r</font> implies <font color="Maroon" title="c10">p</font> <a href="qmax_1.html#R3" title="QMAX_1:pred.3">|-</a> <font color="Maroon" title="c12">r</font> )
 
<b>by </b><i><a class="ref" href="qmax_1.html#T22" target="_self" title="QMAX_1:th.22">Th22</a></i>;<br/>
<b>hence </b><a NAME="E10:41_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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#E2:41_1_2"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="qmax_1.html#E4:41_1_2"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="qmax_1.html#E6:41_1_2"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="qmax_1.html#E8:41_1_2"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="qmax_1.html#T32" target="_self" title="QMAX_1:th.32">Th32</a></i>;<br/>


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

<b>thus </b><a NAME="E3:41_1"/>
 <a href="qmax_1.html#K10" title="QMAX_1:func.10">OrdRel</a> <font color="Maroon" title="c6">Q</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a><br/><b>let </b><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:41_1_3"/><i><font color="Green" title="E30">A8</font></i>: 
( <font color="Maroon" title="c7">x</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; <font color="Maroon" title="c8">y</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>and </b><br/><a NAME="E2:41_1_3"/><i><font color="Green" title="E31">A9</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c9">p</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><b> such that </b><br/><a NAME="E4:41_1_3"/><i><font color="Green" title="E32">A10</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c9">p</font>
 <b>by </b><i><a class="txt" href="qmax_1.html#E1:41_1_3"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="eqrel_1.html#T45" title="EQREL_1:th.45">EQREL_1:45</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">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><b> such that </b><br/><a NAME="E6:41_1_3"/><i><font color="Green" title="E33">A11</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c10">q</font>
 <b>by </b><i><a class="txt" href="qmax_1.html#E1:41_1_3"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="eqrel_1.html#T45" title="EQREL_1:th.45">EQREL_1:45</a></i>;<br/>
<a NAME="E7:41_1_3"/>
( ( <font color="Maroon" title="c9">p</font> <a href="qmax_1.html#R3" title="QMAX_1:pred.3">|-</a> <font color="Maroon" title="c10">q</font> &amp; <font color="Maroon" title="c10">q</font> <a href="qmax_1.html#R3" title="QMAX_1:pred.3">|-</a> <font color="Maroon" title="c9">p</font> implies <font color="Maroon" title="c9">p</font> <a href="qmax_1.html#R4" title="QMAX_1:pred.4">&lt;==&gt;</a> <font color="Maroon" title="c10">q</font> ) &amp; ( <font color="Maroon" title="c9">p</font> <a href="qmax_1.html#R4" title="QMAX_1:pred.4">&lt;==&gt;</a> <font color="Maroon" title="c10">q</font> implies <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">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#D11" target="_self" title="QMAX_1:def.11">Def11</a>, <a class="ref" href="qmax_1.html#D12" target="_self" title="QMAX_1:def.12">Def12</a></i>;<br/>
<b>hence </b><a NAME="E8:41_1_3"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="txt" href="qmax_1.html#E2:41_1_3"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="qmax_1.html#E4:41_1_3"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="qmax_1.html#E6:41_1_3"><i><font color="Green" title="E33">A11</font></i></a>, <a class="ref" href="qmax_1.html#T32" target="_self" title="QMAX_1:th.32">Th32</a>, <a class="ref" href="eqrel_1.html#T44" title="EQREL_1:th.44">EQREL_1:44</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E2:41"/><i><font color="Green" title="E30">A12</font></i>: 
 <a href="qmax_1.html#K11" title="QMAX_1:func.11">InvRel</a> <font color="Maroon" title="c6">Q</font> <a href="qmax_1.html#R1" title="QMAX_1:pred.1">is_an_involution_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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <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>; <i><font color="Red">:: according to </font></i><a class="ref" href="qmax_1.html#D6" target="_self" title="QMAX_1:def.6">QMAX_1:def 6</a><br/>

<b>consider </b><font color="Maroon" title="c8">r</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>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">D</font> =  <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> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>consider </b><font color="Maroon" title="c10">p</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><b> such that </b><br/><a NAME="E4:41_2"/><i><font color="Green" title="E31">A13</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c10">p</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T45" title="EQREL_1:th.45">EQREL_1:45</a></i>;<br/>
<span class="p1">(<span class="default"><a href="qmax_1.html#K11" title="QMAX_1:func.11">InvRel</a> <font color="Maroon" title="c6">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qmax_1.html#K11" title="QMAX_1:func.11">InvRel</a> <font color="Maroon" title="c6">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="qmax_1.html#K11" title="QMAX_1:func.11">InvRel</a> <font color="Maroon" title="c6">Q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<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>,<span class="p2">(<span class="default"><a href="qmax_1.html#K8" title="QMAX_1:func.8">'not'</a> <font color="Maroon" title="c10">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="qmax_1.html#E4:41_2"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="qmax_1.html#D14" target="_self" title="QMAX_1:def.14">Def14</a></i>
<br/>.= 
 <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>,<span class="p1">(<span class="default"><a href="qmax_1.html#K8" title="QMAX_1:func.8">'not'</a> <span class="p2">(<span class="default"><a href="qmax_1.html#K8" title="QMAX_1:func.8">'not'</a> <font color="Maroon" title="c10">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="qmax_1.html#D14" target="_self" title="QMAX_1:def.14">Def14</a></i>
;<br/>
<b>hence </b><a NAME="E6:41_2"/>
<span class="p1">(<span class="default"><a href="qmax_1.html#K11" title="QMAX_1:func.11">InvRel</a> <font color="Maroon" title="c6">Q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qmax_1.html#K11" title="QMAX_1:func.11">InvRel</a> <font color="Maroon" title="c6">Q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="txt" href="qmax_1.html#E4:41_2"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="qmax_1.html#T27" target="_self" title="QMAX_1:th.27">Th27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:41"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <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>  st <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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#K10" title="QMAX_1:func.10">OrdRel</a> <font color="Maroon" title="c6">Q</font> holds <br/><span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qmax_1.html#K11" title="QMAX_1:func.11">InvRel</a> <font color="Maroon" title="c6">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qmax_1.html#K11" title="QMAX_1:func.11">InvRel</a> <font color="Maroon" title="c6">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <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>;<br/>

<b>consider </b><font color="Maroon" title="c9">r</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>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">D</font> =  <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> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>consider </b><font color="Maroon" title="c11">p</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><b> such that </b><br/><a NAME="E4:41_3"/><i><font color="Green" title="E31">A14</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c11">p</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T45" title="EQREL_1:th.45">EQREL_1:45</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">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><b> such that </b><br/><a NAME="E6:41_3"/><i><font color="Green" title="E32">A15</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c12">q</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T45" title="EQREL_1:th.45">EQREL_1:45</a></i>;<br/>
<a NAME="E7:41_3"/><i><font color="Green" title="E33">A16</font></i>: 
( <span class="p1">(<span class="default"><a href="qmax_1.html#K11" title="QMAX_1:func.11">InvRel</a> <font color="Maroon" title="c6">Q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<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="c11">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>,<span class="p1">(<span class="default"><a href="qmax_1.html#K8" title="QMAX_1:func.8">'not'</a> <font color="Maroon" title="c11">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="qmax_1.html#K11" title="QMAX_1:func.11">InvRel</a> <font color="Maroon" title="c6">Q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<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="c12">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>,<span class="p1">(<span class="default"><a href="qmax_1.html#K8" title="QMAX_1:func.8">'not'</a> <font color="Maroon" title="c12">q</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="qmax_1.html#D14" target="_self" title="QMAX_1:def.14">Def14</a></i>;<br/>
<a NAME="E8:41_3"/>
( <font color="Maroon" title="c11">p</font> <a href="qmax_1.html#R3" title="QMAX_1:pred.3">|-</a> <font color="Maroon" title="c12">q</font> implies  <a href="qmax_1.html#K8" title="QMAX_1:func.8">'not'</a> <font color="Maroon" title="c12">q</font> <a href="qmax_1.html#R3" title="QMAX_1:pred.3">|-</a>  <a href="qmax_1.html#K8" title="QMAX_1:func.8">'not'</a> <font color="Maroon" title="c11">p</font> )
 
<b>by </b><i><a class="ref" href="qmax_1.html#T28" target="_self" title="QMAX_1:th.28">Th28</a></i>;<br/>
<b>hence </b><a NAME="E9:41_3"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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#K10" title="QMAX_1:func.10">OrdRel</a> <font color="Maroon" title="c6">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"><span class="p3">(<span class="default"><a href="qmax_1.html#K11" title="QMAX_1:func.11">InvRel</a> <font color="Maroon" title="c6">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qmax_1.html#K11" title="QMAX_1:func.11">InvRel</a> <font color="Maroon" title="c6">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</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#E4:41_3"><i><font color="Green" title="E31">A14</font></i></a>, <a class="txt" href="qmax_1.html#E6:41_3"><i><font color="Green" title="E32">A15</font></i></a>, <a class="txt" href="qmax_1.html#E7:41_3"><i><font color="Green" title="E33">A16</font></i></a>, <a class="ref" href="qmax_1.html#T32" target="_self" title="QMAX_1:th.32">Th32</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:41"/>
 <a href="qmax_1.html#G2" title="QMAX_1:aggr.2">OrthoRelStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p2">(<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></span>)</span>,<span class="p1">(<span class="default"><a href="qmax_1.html#K10" title="QMAX_1:func.10">OrdRel</a> <font color="Maroon" title="c6">Q</font></span>)</span>,<span class="p1">(<span class="default"><a href="qmax_1.html#K11" title="QMAX_1:func.11">InvRel</a> <font color="Maroon" title="c6">Q</font></span>)</span> #) <a href="qmax_1.html#R2" title="QMAX_1:pred.2">is_a_Quantuum_Logic_on</a> 
 <b>by </b><i><a class="txt" href="qmax_1.html#E1:41"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="qmax_1.html#E2:41"><i><font color="Green" title="E30">A12</font></i></a>, <a class="ref" href="qmax_1.html#D7" target="_self" title="QMAX_1:def.7">Def7</a></i>;<br/>


</div>
