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<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> <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="partfun1.html#NM1">PartFunc</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a> , <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:67_1_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">d</font> being   <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a>  holds <br/> ( <font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">d</font>] )
 <b>and </b><br/><a NAME="E3:67_1_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">d</font> being   <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">d</font>)
 <b>from </b><i><a class="ref" href="partfun2.html#S2">PARTFUN2:sch 2</a>();<br/></i>
<a NAME="E4:67_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  <a href="tarski.html#R1">c=</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="ref" href="msualg_1.html#T12">MSUALG_1:12</a></i>;<br/>
<a NAME="E5:67_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> iff <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E6:67_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a> </span>)</span>,<a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/>
<a NAME="E8:67_1_1"/>
<font color="Maroon">f</font> is <a href="comput_1.html#V4" target="_self">homogeneous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">y</font> be   <a href="finseq_1.html#NM1">FinSequence</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="comput_1.html#D4" target="_self">COMPUT_1:def 4</a><br/>

<b>assume </b><a NAME="E1:67_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 ;<br/>

<a NAME="E2:67_1_1_1"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<b>hence </b><a NAME="E3:67_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:67_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is    <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a> </span>)</span>,<a href="numbers.html#K5">NAT</a>  : <font color="Olive">g</font> is <a href="comput_1.html#V4" target="_self">homogeneous</a> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E11:67_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>

<b>let </b><font color="Maroon">p</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E12:67_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a>  <b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> = 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">p'</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i>, </i>
;<br/>


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