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<b>let </b><font color="Maroon">N</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N</font>, <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">N</font>;<br/>



<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:133"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K2">[#]</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="subset_1.html#D4">SUBSET_1:def 4</a></i>;<br/>
<a NAME="E2:133"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N</font>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="subset_1.html#K2">[#]</a> <font color="Maroon">N</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>) <a href="tarski.html#R1">c=</a>  <a href="subset_1.html#K2">[#]</a> <font color="Maroon">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N</font>;<br/>

<b>assume </b><a NAME="E1:133_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="subset_1.html#K2">[#]</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E2:133_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">N</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:133_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">i</font>) <a href="tarski.html#R1">c=</a>  <a href="subset_1.html#K2">[#]</a> <font color="Maroon">N</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">R</font> being   <a href="relset_1.html#NM3">Relation</a> of  <a href="subset_1.html#K2">[#]</a> <font color="Maroon">N</font><b> such that </b><br/><a NAME="E4:133"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N</font>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="subset_1.html#K2">[#]</a> <font color="Maroon">N</font> holds <br/><font color="Maroon">R</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="relset_1.html#S3">RELSET_1:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">R</font> = <font color="Maroon">R</font> as   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">N</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">R</font>
;<br/>

<b>thus </b><a NAME="E6:133"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> holds <br/><font color="Maroon">R</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 <b>by </b><i>, </i>;<br/>


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