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

<b>let </b><font color="Maroon">z1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">z2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:39_1_2"/><i><font color="Green">E53</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">m</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">m</font> holds <br/><font color="Maroon">z1</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K2">IFEQ</a> <font color="Olive">i</font>,2,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Olive">i</font>,3,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font></span>)</span></span>)</span> ) )
 <b>and </b><br/><a NAME="E2:39_1_2"/><i><font color="Green">E55</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">m</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">m</font> holds <br/><font color="Maroon">z2</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K2">IFEQ</a> <font color="Olive">i</font>,2,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Olive">i</font>,3,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font></span>)</span></span>)</span> ) )
 ;<br/>

<i><font color="Green">E56</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">m</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">z1</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#K4">dom</a> <font color="Maroon">z1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<i><font color="Green">E58</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">m</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">z2</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#K4">dom</a> <font color="Maroon">z2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:39_1_2_3"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">m</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_2_3_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3 or ( <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 &amp; <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 3 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:39_1_2_3_1_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:39_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">j</font>,2,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">j</font>,3,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 3
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><a NAME="E3:39_1_2_3_1_1_1"/><b>then </b>
<font color="Maroon">z2</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 3
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:39_1_2_3_1_1_1"/>
<font color="Maroon">z1</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_2_3_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><a NAME="E2:39_1_2_3_1_1_2"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3 implies  <a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">j</font>,2,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">j</font>,3,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_3_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_2_3_1_1_2_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3
 ;<br/>

<b>then </b> <a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">j</font>,2,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">j</font>,3,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> = 
 <a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">j</font>,3,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
<br/>.= 
<font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2
<b>by </b><i><a class="txt" href="idea_1.html#E2"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E3:39_1_2_3_1_1_2_1"/>
 <a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">j</font>,2,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">j</font>,3,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2
 ;<br/>


</div><b>end;</b></div><a NAME="E3:39_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">z2</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:39_1_2_3_1_1_2"/>
<font color="Maroon">z1</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_3_1_1_3"><b>suppose </b></a><a NAME="E1:39_1_2_3_1_1_3"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 &amp; <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 3 )
 ;<br/><div class="add"><a NAME="E2:39_1_2_3_1_1_3"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 &amp; <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 3 implies  <a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">j</font>,2,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">j</font>,3,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_3_1_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_2_3_1_1_3_1"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 &amp; <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 3 )
 ;<br/>

<b>then </b> <a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">j</font>,2,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">j</font>,3,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> = 
 <a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">j</font>,3,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
<br/>.= 
<font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="idea_1.html#E2"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E3:39_1_2_3_1_1_3_1"/>
 <a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">j</font>,2,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">j</font>,3,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E3:39_1_2_3_1_1_3"/><b>then </b>
<font color="Maroon">z2</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:39_1_2_3_1_1_3"/>
<font color="Maroon">z1</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:39_1_2_3"/>
<font color="Maroon">z1</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:39_1_2"/>
<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E1"><i><font color="Green">Lemma32</font></i></a>, , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


</div>
