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

<b>let </b><font color="Maroon">Q</font> be   <a href="qmax_1.html#NM1">Quantum_Mechanics</a>;<br/>

<a NAME="E1:41"/><i><font color="Green">E34</font></i>: 
 <a href="qmax_1.html#K10" target="_self">OrdRel</a> <font color="Maroon">Q</font> <a href="orders_1.html#R2">partially_orders</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">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" target="_self">OrdRel</a> <font color="Maroon">Q</font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D7">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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a><br/>

<b>assume </b><a NAME="E1:41_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qmax_1.html#K5" target="_self">Prop</a> <font color="Maroon">Q</font><b> such that </b><br/><a NAME="E3:41_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span>,<font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T45">EQREL_1:45</a></i>;<br/>
<a NAME="E4:41_1_1"/>
<font color="Maroon">p</font> <a href="qmax_1.html#R3" target="_self">|-</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:41_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="qmax_1.html#K10" target="_self">OrdRel</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="qmax_1.html#D1" target="_self">Def1</a>, </i>;<br/>


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

<b>thus </b><a NAME="E2:41_1"/>
 <a href="qmax_1.html#K10" target="_self">OrdRel</a> <font color="Maroon">Q</font> <a href="relat_2.html#R8">is_transitive_in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:41_1_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:41_1_2"/><i><font color="Green">E48</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="qmax_1.html#K10" target="_self">OrdRel</a> <font color="Maroon">Q</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="qmax_1.html#K10" target="_self">OrdRel</a> <font color="Maroon">Q</font> )
 ;<br/>

<b>consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qmax_1.html#K5" target="_self">Prop</a> <font color="Maroon">Q</font><b> such that </b><br/><a NAME="E4:41_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span>,<font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="qmax_1.html#D1" target="_self">Def1</a>, <a class="ref" href="eqrel_1.html#T45">EQREL_1:45</a></i>;<br/>
<b>consider </b><font color="Maroon">q</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qmax_1.html#K5" target="_self">Prop</a> <font color="Maroon">Q</font><b> such that </b><br/><a NAME="E6:41_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span>,<font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="qmax_1.html#D1" target="_self">Def1</a>, <a class="ref" href="eqrel_1.html#T45">EQREL_1:45</a></i>;<br/>
<b>consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qmax_1.html#K5" target="_self">Prop</a> <font color="Maroon">Q</font><b> such that </b><br/><a NAME="E8:41_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="qmax_1.html#D1" target="_self">Def1</a>, <a class="ref" href="eqrel_1.html#T45">EQREL_1:45</a></i>;<br/>
<a NAME="E9:41_1_2"/>
( <font color="Maroon">p</font> <a href="qmax_1.html#R3" target="_self">|-</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="qmax_1.html#R3" target="_self">|-</a> <font color="Maroon">r</font> implies <font color="Maroon">p</font> <a href="qmax_1.html#R3" target="_self">|-</a> <font color="Maroon">r</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:41_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="qmax_1.html#K10" target="_self">OrdRel</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="txt" href="qmax_1.html#E1:41"><i><font color="Green">E34</font></i></a>, , <a class="txt" href="qmax_1.html#E1:41_1_2"><i><font color="Green">E36</font></i></a>, </i>;<br/>


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

<b>thus </b><a NAME="E3:41_1"/>
 <a href="qmax_1.html#K10" target="_self">OrdRel</a> <font color="Maroon">Q</font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D4">RELAT_2:def 4</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:41_1_3"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:41_1_3"/><i><font color="Green">E54</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="qmax_1.html#K10" target="_self">OrdRel</a> <font color="Maroon">Q</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="qmax_1.html#K10" target="_self">OrdRel</a> <font color="Maroon">Q</font> )
 ;<br/>

<b>consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qmax_1.html#K5" target="_self">Prop</a> <font color="Maroon">Q</font><b> such that </b><br/><a NAME="E4:41_1_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span>,<font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="qmax_1.html#D1" target="_self">Def1</a>, <a class="ref" href="eqrel_1.html#T45">EQREL_1:45</a></i>;<br/>
<b>consider </b><font color="Maroon">q</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qmax_1.html#K5" target="_self">Prop</a> <font color="Maroon">Q</font><b> such that </b><br/><a NAME="E6:41_1_3"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span>,<font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="qmax_1.html#D1" target="_self">Def1</a>, <a class="ref" href="eqrel_1.html#T45">EQREL_1:45</a></i>;<br/>
<a NAME="E7:41_1_3"/>
( ( <font color="Maroon">p</font> <a href="qmax_1.html#R3" target="_self">|-</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="qmax_1.html#R3" target="_self">|-</a> <font color="Maroon">p</font> implies <font color="Maroon">p</font> <a href="qmax_1.html#R4" target="_self">&lt;==&gt;</a> <font color="Maroon">q</font> ) &amp; ( <font color="Maroon">p</font> <a href="qmax_1.html#R4" target="_self">&lt;==&gt;</a> <font color="Maroon">q</font> implies <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font> ) )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:41_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="txt" href="qmax_1.html#E1:41"><i><font color="Green">E34</font></i></a>, , , <a class="ref" href="eqrel_1.html#T44">EQREL_1:44</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E2:41"/><i><font color="Green">E57</font></i>: 
 <a href="qmax_1.html#K11" target="_self">InvRel</a> <font color="Maroon">Q</font> <a href="qmax_1.html#R1" target="_self">is_an_involution_in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">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">x</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="qmax_1.html#D6" target="_self">QMAX_1:def 6</a><br/>

<b>consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qmax_1.html#K5" target="_self">Prop</a> <font color="Maroon">Q</font>;<br/>
<b>reconsider </b><font color="Maroon">D</font> =  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qmax_1.html#K5" target="_self">Prop</a> <font color="Maroon">Q</font><b> such that </b><br/><a NAME="E4:41_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span>,<font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T45">EQREL_1:45</a></i>;<br/>
<span class="p1">(<span class="default"><a href="qmax_1.html#K11" target="_self">InvRel</a> <font color="Maroon">Q</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qmax_1.html#K11" target="_self">InvRel</a> <font color="Maroon">Q</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="qmax_1.html#K11" target="_self">InvRel</a> <font color="Maroon">Q</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span>,<span class="p2">(<span class="default"><a href="qmax_1.html#K8" target="_self">'not'</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span>,<span class="p1">(<span class="default"><a href="qmax_1.html#K8" target="_self">'not'</a> <span class="p2">(<span class="default"><a href="qmax_1.html#K8" target="_self">'not'</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E6:41_2"/>
<span class="p1">(<span class="default"><a href="qmax_1.html#K11" target="_self">InvRel</a> <font color="Maroon">Q</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qmax_1.html#K11" target="_self">InvRel</a> <font color="Maroon">Q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:41"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span>  st <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="qmax_1.html#K10" target="_self">OrdRel</a> <font color="Maroon">Q</font> holds <br/><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qmax_1.html#K11" target="_self">InvRel</a> <font color="Maroon">Q</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qmax_1.html#K11" target="_self">InvRel</a> <font color="Maroon">Q</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="qmax_1.html#K10" target="_self">OrdRel</a> <font color="Maroon">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">x</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span>, <font color="Maroon">y</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qmax_1.html#K5" target="_self">Prop</a> <font color="Maroon">Q</font>;<br/>
<b>reconsider </b><font color="Maroon">D</font> =  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qmax_1.html#K5" target="_self">Prop</a> <font color="Maroon">Q</font><b> such that </b><br/><a NAME="E4:41_3"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span>,<font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T45">EQREL_1:45</a></i>;<br/>
<b>consider </b><font color="Maroon">q</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qmax_1.html#K5" target="_self">Prop</a> <font color="Maroon">Q</font><b> such that </b><br/><a NAME="E6:41_3"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span>,<font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T45">EQREL_1:45</a></i>;<br/>
<a NAME="E7:41_3"/><i><font color="Green">E117</font></i>: 
( <span class="p1">(<span class="default"><a href="qmax_1.html#K11" target="_self">InvRel</a> <font color="Maroon">Q</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span>,<font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span>,<span class="p1">(<span class="default"><a href="qmax_1.html#K8" target="_self">'not'</a> <font color="Maroon">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="qmax_1.html#K11" target="_self">InvRel</a> <font color="Maroon">Q</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span>,<font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span>,<span class="p1">(<span class="default"><a href="qmax_1.html#K8" target="_self">'not'</a> <font color="Maroon">q</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E8:41_3"/>
( <font color="Maroon">p</font> <a href="qmax_1.html#R3" target="_self">|-</a> <font color="Maroon">q</font> implies  <a href="qmax_1.html#K8" target="_self">'not'</a> <font color="Maroon">q</font> <a href="qmax_1.html#R3" target="_self">|-</a>  <a href="qmax_1.html#K8" target="_self">'not'</a> <font color="Maroon">p</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:41_3"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="qmax_1.html#K10" target="_self">OrdRel</a> <font color="Maroon">Q</font> implies <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qmax_1.html#K11" target="_self">InvRel</a> <font color="Maroon">Q</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qmax_1.html#K11" target="_self">InvRel</a> <font color="Maroon">Q</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="qmax_1.html#K10" target="_self">OrdRel</a> <font color="Maroon">Q</font> )
 <b>by </b><i>, <a class="txt" href="qmax_1.html#E1:41"><i><font color="Green">E34</font></i></a>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:41"/>
 <a href="qmax_1.html#G2" target="_self">POI_Str</a>(# <span class="p1">(<span class="default"><a href="qmax_1.html#K10" target="_self">OrdRel</a> <font color="Maroon">Q</font></span>)</span>,<span class="p1">(<span class="default"><a href="qmax_1.html#K11" target="_self">InvRel</a> <font color="Maroon">Q</font></span>)</span> #) <a href="qmax_1.html#R2" target="_self">is_a_Quantuum_Logic_on</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="qmax_1.html#K9" target="_self">PropRel</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="qmax_1.html#D1" target="_self">Def1</a>, </i>;<br/>


</div>
