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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="funct_1.html#NM1">Function</a> =  <a href="pnproc_1.html#K9" target="_self">fire</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E2:60_1_1"/><i><font color="Green">E29</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E3:60_1_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E4:60_1_1"/><i><font color="Green">E35</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:60_1_1"/>
<font color="Maroon">F</font> is <a href="funcop_1.html#V1">Function-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:60_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E3:60_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="pnproc_1.html#K9" target="_self">fire</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pnproc_1.html#T4" target="_self">Th4</a>, , , </i>;<br/>
<b>hence </b><a NAME="E4:60_1_1_1"/>
<font color="Maroon">x</font> <a href="pnproc_1.html#NK1" target="_self">multitude_of</a>  is    <a href="hidden.html#M1">set</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as  <a href="funcop_1.html#V1">Function-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> ;<br/>
<b>take </b><font color="Maroon">f</font> =  <a href="funct_7.html#K4">compose</a> <font color="Maroon">F</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">P</font>,<a href="numbers.html#K5">NAT</a> </span>)</span>;<br/>

<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E7:60_1_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="funct_7.html#K4">compose</a> <font color="Maroon">F</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">P</font>,<a href="numbers.html#K5">NAT</a> </span>)</span>
 ;<br/>

<b>thus </b><a NAME="E8:60_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">C</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="pnproc_1.html#K9" target="_self">fire</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="pnproc_1.html#T4" target="_self">Th4</a>, , </i>;<br/>


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