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

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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_Quantum_Logic</a> </span><br/>

<a NAME="E1:62"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
 <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="62_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <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> or  not <font color="Olive">b<sub>1</sub></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> or  not <font color="Olive">b<sub>2</sub></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> or  not <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="Olive">b<sub>1</sub></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> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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> or <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="Olive">b<sub>2</sub></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> )</span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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> or  not <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> or  not <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> or  not <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> or  not <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> or <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> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:62_1"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:62_1"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:62_1"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:62_1"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
( <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E6:62_1"/><span class="lab"><font color="Green" title="E32">A6</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="eqrel_1.html#T45" title="EQREL_1:th.45">EQREL_1:45</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">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><span class="kw"> such that </span><br/><a NAME="E8:62_1"/><span class="lab"><font color="Green" title="E33">A7</font></span>: 
<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="c11">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="eqrel_1.html#T45" title="EQREL_1:th.45">EQREL_1:45</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E10:62_1"/><span class="lab"><font color="Green" title="E34">A8</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="eqrel_1.html#T45" title="EQREL_1:th.45">EQREL_1:45</a></span>;<br/>
<a NAME="E11:62_1"/>
( <font color="Maroon" title="c10">p</font> <a href="qmax_1.html#R3" title="QMAX_1:pred.3">|-</a> <font color="Maroon" title="c12">q</font> &amp; <font color="Maroon" title="c12">q</font> <a href="qmax_1.html#R3" title="QMAX_1:pred.3">|-</a> <font color="Maroon" title="c11">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="c11">r</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qmax_1.html#T22" target="_self" title="QMAX_1:th.22">Th22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:62_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="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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E6:62_1"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E10:62_1"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="txt" href="#E8:62_1"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="qmax_1.html#T32" target="_self" title="QMAX_1:th.32">Th32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:62"/><span class="lab"><font color="Green" title="E28">A9</font></span>: 
 <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="62_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <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> or  not <font color="Olive">b<sub>1</sub></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> or  not <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="Olive">b<sub>1</sub></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> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></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> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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> or  not <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> or  not <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> or  not <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> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:62_2"/><span class="lab"><font color="Green" title="E29">A10</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:62_2"/><span class="lab"><font color="Green" title="E30">A11</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:62_2"/><span class="lab"><font color="Green" title="E31">A12</font></span>: 
( <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font></span><br/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:62_2"/><span class="lab"><font color="Green" title="E32">A13</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="eqrel_1.html#T45" title="EQREL_1:th.45">EQREL_1:45</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E7:62_2"/><span class="lab"><font color="Green" title="E33">A14</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_2"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="eqrel_1.html#T45" title="EQREL_1:th.45">EQREL_1:45</a></span>;<br/>
<a NAME="E8:62_2"/><span class="lab"><font color="Green" title="E34">A15</font></span>: 
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qmax_1.html#D12" target="_self" title="QMAX_1:def.12">Def12</a></span>;<br/>
<a NAME="E9:62_2"/>
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qmax_1.html#D11" target="_self" title="QMAX_1:def.11">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:62_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_2"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E5:62_2"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E7:62_2"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="txt" href="#E8:62_2"><span class="lab"><font color="Green" title="E34">A15</font></span></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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:62"/><span class="lab"><font color="Green" title="E29">A16</font></span>: 
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="62_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c7">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/>
<span class="kw">let </span><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">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="c9">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="c8">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> )</span><br/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:62_3"/><span class="lab"><font color="Green" title="E30">A17</font></span>: 
<font color="Maroon" title="c8">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T45" title="EQREL_1:th.45">EQREL_1:45</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:62_3"/><span class="lab"><font color="Green" title="E31">A18</font></span>: 
<font color="Maroon" title="c9">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T45" title="EQREL_1:th.45">EQREL_1:45</a></span>;<br/>
<a NAME="E6:62_3"/><span class="lab"><font color="Green" title="E32">A19</font></span>: 
( <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> implies  <a href="qmax_1.html#K8" title="QMAX_1:func.8">'not'</a> <font color="Maroon" title="c11">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="c10">p</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qmax_1.html#T28" target="_self" title="QMAX_1:th.28">Th28</a></span>;<br/>
<a NAME="E7:62_3"/>
( <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="c10">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="c10">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="c11">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="c11">q</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qmax_1.html#D14" target="_self" title="QMAX_1:def.14">Def14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:62_3"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">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="c9">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="c8">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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_3"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="txt" href="#E5:62_3"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="txt" href="#E6:62_3"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="ref" href="qmax_1.html#T32" target="_self" title="QMAX_1:th.32">Th32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:62"/><span class="lab"><font color="Green" title="E30">A20</font></span>: 
 <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</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c7">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/>
<span class="kw">let </span><font color="Maroon" title="c8">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>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="qmax_1.html#D6" target="_self" title="QMAX_1:def.6">QMAX_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="c8">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font></span><br/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:62_4"/><span class="lab"><font color="Green" title="E31">A21</font></span>: 
<font color="Maroon" title="c8">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T45" title="EQREL_1:th.45">EQREL_1:45</a></span>;<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="c8">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="c9">p</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_4"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="ref" href="qmax_1.html#D14" target="_self" title="QMAX_1:def.14">Def14</a></span>
<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="c9">p</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="qmax_1.html#D14" target="_self" title="QMAX_1:def.14">Def14</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:62_4"/>
<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="c8">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_4"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="ref" href="qmax_1.html#T27" target="_self" title="QMAX_1:th.27">Th27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:62"/>
 <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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> or <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:62_5"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:62_5"/><span class="lab"><font color="Green" title="E31">A22</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T45" title="EQREL_1:th.45">EQREL_1:45</a></span>;<br/>
<a NAME="E4:62_5"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qmax_1.html#T21" target="_self" title="QMAX_1:th.21">Th21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:62_5"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_5"><span class="lab"><font color="Green" title="E31">A22</font></span></a>, <a class="ref" href="qmax_1.html#T32" target="_self" title="QMAX_1:th.32">Th32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:62"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="ref" href="orders_1.html#D7" title="ORDERS_1:def.7">ORDERS_1:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:62"/>
 <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_Quantum_Logic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62"><span class="lab"><font color="Green" title="E30">A20</font></span></a>, <a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="ref" href="qmax_1.html#D7" target="_self" title="QMAX_1:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
