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

<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>;<br/><b>let </b><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</font> be    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<a NAME="E1:61"/><i><font color="Green" title="E60">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f2</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c3">f2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="nat_4.html#D1" target="_self" title="NAT_4:def.1">Def1</a></i>;<br/>
<i><font color="Green" title="E61">A2</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nat_4.html#D1" target="_self" title="NAT_4:def.1">Def1</a></i>
;<br/>
<i><font color="Green" title="E62">A3</font></i>:  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c2">f1</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E1:61"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="nat_4.html#D1" target="_self" title="NAT_4:def.1">Def1</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E4:61"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">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" title="c4">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:61_3"/><i><font color="Green" title="E63">A4</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:61_3"/><b>then </b><i><font color="Green" title="E64">A5</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c3">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E2:61"><i><font color="Green" title="E61">A2</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f1</font> or  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f2</font> &amp; <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> ) )
 </span><b>by </b><i><a class="txt" href="nat_4.html#E2:61_3"><i><font color="Green" title="E64">A5</font></i></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">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" title="E65">A6</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f1</font>
 ;<br/><div class="add"><i><font color="Green" title="E66">A7</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f1</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f1</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c2">f1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nat_4.html#D1" target="_self" title="NAT_4:def.1">Def1</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c2">f1</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">k</font> = 
<font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E1:61_3"><i><font color="Green" title="E63">A4</font></i></a>, <a class="ref" href="nat_4.html#D1" target="_self" title="NAT_4:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E1:61_3_1_1"><i><font color="Green" title="E65">A6</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">k</font>
<b>by </b><i><a class="txt" href="nat_4.html#E1:61_3_1_1"><i><font color="Green" title="E65">A6</font></i></a>, <a class="txt" href="nat_4.html#E2:61_3_1_1"><i><font color="Green" title="E66">A7</font></i></a>, <a class="ref" href="nat_4.html#D1" target="_self" title="NAT_4:def.1">Def1</a></i>
;<br/><b>hence </b><a NAME="E4:61_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:61_3_1_1"><i><font color="Green" title="E65">A6</font></i></a>, <a class="txt" href="nat_4.html#E2:61_3_1_1"><i><font color="Green" title="E66">A7</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">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"/>
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f2</font> &amp; <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:61_3_1_2"/><i><font color="Green" title="E65">A8</font></i>: 
( <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f2</font> &amp; <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font> )
 ;<br/><a NAME="E4:61_3_1_2"/><i><font color="Green" title="E66">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E3:61_3_1_2"><i><font color="Green" title="E65">A8</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><i><font color="Green" title="E67">A10</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f2</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nat_4.html#D1" target="_self" title="NAT_4:def.1">Def1</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c3">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/><i><font color="Green" title="E68">A11</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c2">f1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E3:61_3_1_2"><i><font color="Green" title="E65">A8</font></i></a>, <a class="ref" href="nat_4.html#D1" target="_self" title="NAT_4:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
<b>by </b><i><a class="txt" href="nat_4.html#E3:61_3_1_2"><i><font color="Green" title="E65">A8</font></i></a>, <a class="txt" href="nat_4.html#E5:61_3_1_2"><i><font color="Green" title="E67">A10</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">k</font> = 
<font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E1:61_3"><i><font color="Green" title="E63">A4</font></i></a>, <a class="txt" href="nat_4.html#E4:61_3_1_2"><i><font color="Green" title="E66">A9</font></i></a>, <a class="ref" href="nat_4.html#D1" target="_self" title="NAT_4:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
<b>by </b><i><a class="txt" href="nat_4.html#E3:61_3_1_2"><i><font color="Green" title="E65">A8</font></i></a>, <a class="txt" href="nat_4.html#E5:61_3_1_2"><i><font color="Green" title="E67">A10</font></i></a>, <a class="ref" href="nat_4.html#D1" target="_self" title="NAT_4:def.1">Def1</a></i>
;<br/><b>hence </b><a NAME="E8:61_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E6:61_3_1_2"><i><font color="Green" title="E68">A11</font></i></a></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" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <font color="Maroon" title="c3">f2</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E2:61"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="nat_4.html#E3:61"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/>


</div>
