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<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">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="ordinal3.html#K5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font></span>)</span></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:3_1_1"/><i><font color="Green">E13</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">psi</font></span>)</span>
 <b>and </b><br/><a NAME="E3:3_1_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">x</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">psi</font></span>)</span> holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>) ) )
 <b>from </b><i><a class="ref" href="finset_1.html#S1">FINSET_1:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="ordinal1.html#NM3">T-Sequence</a> <b>by </b><i>, <a class="ref" href="ordinal1.html#D7">ORDINAL1:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E5:3_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">psi</font></span>)</span>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E6:3_1_1"/>
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:3_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font>
 ;<br/>

<a NAME="E2:3_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> <a href="ordinal1.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="ordinal3.html#T27">ORDINAL3:27</a></i>;<br/>
<b>hence </b><a NAME="E3:3_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E7:3_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">psi</font>
 ;<br/>

<a NAME="E8:3_1_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="ordinal2.html#T49">ORDINAL2:49</a></i>;<br/>
<a NAME="E9:3_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> <a href="ordinal1.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T27">ORDINAL3:27</a></i>;<br/>
<a NAME="E10:3_1_1"/><b>then </b>
not <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/>
<a NAME="E11:3_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal3.html#K5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal3.html#K5">-^</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , , <a class="ref" href="ordinal3.html#T65">ORDINAL3:65</a></i>;<br/>
<b>hence </b><a NAME="E12:3_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font>
 ;<br/>


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