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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> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp;  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<sub>1</sub></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">a<sub>2</sub></font> );<br/>
<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:30_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 ;<br/><a NAME="E2:30_1_1_1"/>
<span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">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>,0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/><b>hence </b><a NAME="E3:30_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">x</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E2:30_1_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>]
 
;<br/>
<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2"><b>now </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>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:30_1_1_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">z</font>] )
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:30_1_1_2"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">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="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/><b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:30_1_1_2"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</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="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/><a NAME="E6:30_1_1_2"/><i><font color="Green">E59</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T67">FUNCT_2:67</a></i>;<br/><a NAME="E7:30_1_1_2"/><i><font color="Green">E60</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">l</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T67">FUNCT_2:67</a></i>;<br/><a NAME="E8:30_1_1_2"/><i><font color="Green">E61</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T67">FUNCT_2:67</a></i>;<br/><a NAME="E9:30_1_1_2"/><i><font color="Green">E62</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T67">FUNCT_2:67</a></i>;<br/><span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79">FUNCT_7:79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">l</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79">FUNCT_7:79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79">FUNCT_7:79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79">FUNCT_7:79</a></i>
;<br/><b>hence </b><a NAME="E11:30_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">z</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">EqR</font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E5:30_1_1"/><i><font color="Green">E64</font></i>: 
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">EqR</font> iff ( <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> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] ) )
 <b>from </b><i><a class="ref" href="eqrel_1.html#S1">EQREL_1:sch 1</a>(, , );<br/></i>
<b>take </b>
<font color="Maroon">EqR</font>
;<br/>

<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>assume </b><a NAME="E6:30_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> )
 ;<br/>

<b>hence </b><a NAME="E7:30_1_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">EqR</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="Maroon">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="Maroon">y</font> )
 <b>by </b><i/>;<br/>


</div>
