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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">p</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>  st <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">p</font>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K10" target="_self">-'</a> <font color="Olive">p</font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>;<br/>
<a NAME="E1:149_1_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">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> <span class="p2">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <font color="Maroon">b</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive">d</font> being   <a href="finseq_2.html#M2">Element</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">d</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="149_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:149_1_1_1"/>
<font color="Maroon">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> <span class="p2">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">p</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="polynom1.html#T6" target="_self">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">p</font>, <font color="Maroon">b2</font> = <font color="Maroon">b</font> <a href="polynom1.html#K10" target="_self">-'</a> <font color="Maroon">p</font> as    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="polynom1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E4:149_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K10" target="_self">-'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">d</font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<b>take </b>
<font color="Maroon">d</font>
;<br/>

<b>thus </b><a NAME="E6:149_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">d</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:149_1_1"/><i><font color="Green">E42</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> <span class="p1">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E4:149_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</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> <span class="p2">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <font color="Maroon">b</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">f</font> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="goboard1.html#S1">GOBOARD1:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E5:149_1_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#T14" target="_self">Th14</a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E6:149_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E7:149_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E8:149_1_1"/>
<font color="Maroon">i</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> <span class="p2">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#T14" target="_self">Th14</a>, <a class="ref" href="polynom1.html#T16" target="_self">Th16</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E9:149_1_1"/>
<font color="Maroon">f</font> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K10" target="_self">-'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="polynom1.html#T15" target="_self">Th15</a>, <a class="ref" href="polynom1.html#T17" target="_self">Th17</a></i>;<br/>


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