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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qmax_1.html#K5" target="_self">Prop</a> <font color="Maroon">c<sub>6</sub></font> st <font color="Maroon">a<sub>1</sub></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">c<sub>6</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="qmax_1.html#K8" target="_self">'not'</a> <font color="Olive">b<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:40_1_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></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">c<sub>6</sub></font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:40_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>7</sub></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">c<sub>6</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qmax_1.html#K5" target="_self">Prop</a> <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E3:40_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>7</sub></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">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T45">EQREL_1:45</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <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">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="qmax_1.html#K8" target="_self">'not'</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E5:40_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="qmax_1.html#K5" target="_self">Prop</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> ;<br/>
<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qmax_1.html#K5" target="_self">Prop</a> <font color="Maroon">c<sub>6</sub></font>;<br/>

<b>assume </b><a NAME="E7:40_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>7</sub></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">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="qmax_1.html#K5" target="_self">Prop</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> ;<br/>
<a NAME="E9:40_1_1_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="qmax_1.html#K8" target="_self">'not'</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="qmax_1.html#E3:40_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="qmax_1.html#E7:40_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/>
<a NAME="E10:40_1_1_1"/><b>then </b>
 <a href="qmax_1.html#K8" target="_self">'not'</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="qmax_1.html#E1:40_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="qmax_1.html#E5:40_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="qmax_1.html#T33" target="_self">Th33</a></i>;<br/>
<b>hence </b><a NAME="E11:40_1_1_1"/>
<font color="Maroon">c<sub>9</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="qmax_1.html#K8" target="_self">'not'</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="funct_2.html#NM1">Function</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">c<sub>6</sub></font></span>)</span>, <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">c<sub>6</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:40_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="qmax_1.html#E1:40_1_1"><i><font color="Green">E30</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qmax_1.html#K5" target="_self">Prop</a> <font color="Maroon">c<sub>6</sub></font>;<br/>

<a NAME="E4:40_1_1"/>
 <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">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></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">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:40_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <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">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="qmax_1.html#K8" target="_self">'not'</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="qmax_1.html#E3:40_1_1"><i><font color="Green">E31</font></i></a></i>;<br/>


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