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

<b>let </b><font color="Maroon">R1</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of  <a href="qmax_1.html#K5" target="_self">Prop</a> <font color="Maroon">Q</font>, <font color="Maroon">R2</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of  <a href="qmax_1.html#K5" target="_self">Prop</a> <font color="Maroon">Q</font>;<br/>

<b>assume </b><a NAME="E1:34_1_2"/><i><font color="Green">E56</font></i>: 
( ( 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> holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">p</font>,<font color="Olive">q</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">p</font> <a href="qmax_1.html#R4" target="_self">&lt;==&gt;</a> <font color="Olive">q</font> ) ) &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> holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">p</font>,<font color="Olive">q</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">p</font> <a href="qmax_1.html#R4" target="_self">&lt;==&gt;</a> <font color="Olive">q</font> ) ) )
 ;<br/>

<div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <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>let </b><font color="Maroon">q</font> be    <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/><a NAME="E1:34_1_2_1"/>
( ( <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> <font color="Maroon">R1</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> <font color="Maroon">R1</font> ) &amp; ( <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> <font color="Maroon">R2</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> <font color="Maroon">R2</font> ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:34_1_2_1"/>
( <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> <font color="Maroon">R1</font> iff <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> <font color="Maroon">R2</font> )
 ;<br/></div><b>end;</b></div>
<a NAME="E3:34_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="34_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="E1:34_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="34_1_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1_2_2_1"/><i><font color="Green">E96</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="E2:34_1_2_2_1"/><b>then </b>
( <font color="Maroon">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="qmax_1.html#K5" target="_self">Prop</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">y</font> is    <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>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E3:34_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="E2:34_1_2_2"/><i><font color="Green">E97</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="E3:34_1_2_2"/><b>then </b>
( <font color="Maroon">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="qmax_1.html#K5" target="_self">Prop</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">y</font> is    <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>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E4:34_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="E4:34_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/>


</div>
