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<b>let </b><font color="Maroon">IT1</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">E</font>, <font color="Maroon">IT2</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">E</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:30_1_2"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> 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">IT1</font> iff  ex <font color="Olive">k</font>, <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> )
 <b>and </b><br/><a NAME="E2:30_1_2"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> 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">IT2</font> iff  ex <font color="Olive">k</font>, <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> )
 ;<br/>

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a><br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:30_1_2_1"/><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">IT1</font>
 ;<br/><a NAME="E2:30_1_2_1"/><b>then </b><i><font color="Green">E71</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:30_1_2_1"/><b>then </b>
 ex <font color="Olive">k</font>, <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:30_1_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">IT2</font>
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:30_1_2"/><i><font color="Green">E72</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">IT2</font>
 ;<br/>

<a NAME="E5:30_1_2"/><b>then </b><i><font color="Green">E73</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:30_1_2"/><b>then </b>
 ex <font color="Olive">k</font>, <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:30_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">IT1</font>
 <b>by </b><i>, </i>;<br/>


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