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<div class="add">

<b>let </b><font color="Maroon">S</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">R</font> be  <a href="partfun1.html#V1">total</a>  <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">S</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">s0</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>;<br/>





<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>) <b>-&gt; </b> non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  = <font color="Maroon">S</font>;<br/>
<a NAME="E1:82"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> holds  <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">s</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">s</font>) is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><div class="add">

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

<b>set </b><font color="Maroon">X</font> = <font color="Maroon">R</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E1:82_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">R</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T144">RELAT_1:144</a></i>;<br/>
<a NAME="E2:82_1"/><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E3:82_1"/><b>then </b>
<font color="Maroon">R</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E4:82_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<b>hence </b><a NAME="E5:82_1"/>
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s</font>) is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">S</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:82"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s0</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>) ) ) )
 <b>from </b><i><a class="ref" href="modelc_1.html#S1" target="_self">MODELC_1:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E4:82"/>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s0</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) )
 <b>by </b><i/>;<br/>


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