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<div class="add">

<b>let </b><font color="Maroon">R1</font> be   <a href="relset_1.html#NM3">Relation</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">R2</font> be   <a href="relset_1.html#NM3">Relation</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>assume </b><b>that </b><br/><a NAME="E1:36_1_2"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">B</font>, <font color="Olive">C</font> being  <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">Q</font></span>)</span> holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">B</font>,<font color="Olive">C</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> iff ( <font color="Olive">B</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="Olive">C</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; ( for <font color="Olive">p</font>, <font color="Olive">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>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> holds <br/><font color="Olive">p</font> <a href="qmax_1.html#R3" target="_self">|-</a> <font color="Olive">q</font> ) ) )
 <b>and </b><br/><a NAME="E2:36_1_2"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">B</font>, <font color="Olive">C</font> being  <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">Q</font></span>)</span> holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">B</font>,<font color="Olive">C</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> iff ( <font color="Olive">B</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="Olive">C</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; ( for <font color="Olive">p</font>, <font color="Olive">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>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> holds <br/><font color="Olive">p</font> <a href="qmax_1.html#R3" target="_self">|-</a> <font color="Olive">q</font> ) ) )
 ;<br/>

<div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">B</font> be   <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">Q</font></span>)</span>;<br/><b>let </b><font color="Maroon">C</font> be   <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">Q</font></span>)</span>;<br/><a NAME="E1:36_1_2_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> iff ( <font color="Maroon">B</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">C</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; ( for <font color="Olive">p</font>, <font color="Olive">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>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/><font color="Olive">p</font> <a href="qmax_1.html#R3" target="_self">|-</a> <font color="Olive">q</font> ) ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:36_1_2_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> iff <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E4:36_1_2"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2_2"><b>proof </b></a><div class="add">

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



<b>thus </b><a NAME="E1:36_1_2_2"/>
( <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> <font color="Maroon">R1</font> implies <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> <font color="Maroon">R2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_1_2_2_1"/><i><font color="Green">E49</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> <font color="Maroon">R1</font>
 ;<br/>

<a NAME="E2:36_1_2_2_1"/><b>then </b>
( <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>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E3:36_1_2_2_1"/>
<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> <font color="Maroon">R2</font>
 <b>by </b><i>, </i>;<br/>


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

<b>assume </b><a NAME="E2:36_1_2_2"/><i><font color="Green">E50</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> <font color="Maroon">R2</font>
 ;<br/>

<a NAME="E3:36_1_2_2"/><b>then </b>
( <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>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E4:36_1_2_2"/>
<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> <font color="Maroon">R1</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:36_1_2"/>
<font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R2</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a></i>;<br/>


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