<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/><b>let </b><font color="Maroon">f1</font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">f2</font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<a NAME="E1:61"/><i><font color="Green">E38</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> &amp; ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f2</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="polynom1.html#K8">.</a> <font color="Olive">i</font></span>)</span> ) )
 
<b>by </b><i/>;<br/>
<i><font color="Green">E39</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">f2</font></span>)</span></span>)</span> = 
 <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"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <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"><font color="Maroon">f1</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">f2</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<i><font color="Green">E40</font></i>:  <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"><font color="Maroon">f1</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">f2</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <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"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f1</font></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f1</font></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E4:61"/>
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="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">f2</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f1</font></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_3"><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="E1:61_3"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">f2</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:61_3"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_3_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> or  ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Olive">n</font> ) )
 </span><b>by </b><i>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_3_1_1"><b>suppose </b></a><a NAME="E1:61_3_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 ;<br/><div class="add"><i><font color="Green">E53</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> = 
 <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">f1</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <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"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f1</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f1</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">f2</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f1</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , </i>
;<br/><b>hence </b><a NAME="E4:61_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f1</font></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_3_1_2"><b>suppose </b></a><a NAME="E1:61_3_1_2"/><i><font color="Green">E54</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Olive">n</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:61_3_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/><a NAME="E4:61_3_1_2"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">f2</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><i><font color="Green">E58</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> = 
 <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">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <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"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f2</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><i><font color="Green">E60</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f1</font></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f1</font></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f2</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a>, , </i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f2</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">n</font>
<b>by </b><i>, , </i>
;<br/><b>hence </b><a NAME="E8:61_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f1</font></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:61"/>
<font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f1</font></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a>, </i>;<br/>


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