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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 3 ) &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 2 ) &amp; ( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 &amp; $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 ) );<br/>
<a NAME="E1:39_1_1"/><i><font color="Green" title="E30">A1</font></i>: 
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#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="c1">m</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:39_1_1_1"/>
<font color="Maroon" title="c2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c1">m</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">k</font> = <font color="Maroon" title="c2">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_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_1_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or ( <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 &amp; <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_1"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><b>take </b>
<font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 3
;<br/><b>thus </b><a NAME="E2:39_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 3]
 <b>by </b><i><a class="txt" href="idea_1.html#E1:39_1_1_1_1_1"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_2"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><div class="add"><b>take </b>
<font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 2
;<br/><b>thus </b><a NAME="E2:39_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 2]
 <b>by </b><i><a class="txt" href="idea_1.html#E1:39_1_1_1_1_2"><i><font color="Green" title="E31">A3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_3"/><i><font color="Green" title="E31">A4</font></i>: 
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 &amp; <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 )
 ;<br/><div class="add"><b>take </b>
<font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c3">k</font>
;<br/><b>thus </b><a NAME="E2:39_1_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c3">k</font>]
 <b>by </b><i><a class="txt" href="idea_1.html#E1:39_1_1_1_1_3"><i><font color="Green" title="E31">A4</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c2">z</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:39_1_1"/><i><font color="Green" title="E31">A5</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c1">m</font></span>)</span>
 <b>and </b><br/><a NAME="E4:39_1_1"/><i><font color="Green" title="E32">A6</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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#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="c1">m</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="finseq_1.html#S5" title="FINSEQ_1:sch.5">FINSEQ_1:sch 5</a>(<a class="txt" href="idea_1.html#E1:39_1_1"><i><font color="Green" title="E30">A1</font></i></a>);<br/></i>
<a NAME="E5:39_1_1"/><i><font color="Green" title="E33">A7</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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#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="c1">m</font></span>)</span> holds <br/> <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Olive" title="b1">i</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Olive" title="b1">i</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:39_1_1_2"/><i><font color="Green" title="E34">A8</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c1">m</font></span>)</span>
 ;<br/>

<a NAME="E2:39_1_1_2"/><i><font color="Green" title="E35">A9</font></i>: 
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies  <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c3">i</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c3">i</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 3 )
 
<b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/>
<a NAME="E3:39_1_1_2"/><i><font color="Green" title="E36">A10</font></i>: 
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 implies  <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c3">i</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c3">i</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 2 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_1_2_1"/><i><font color="Green" title="E37">A11</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/>

<b>then </b> <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c3">i</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c3">i</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> = 
 <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c3">i</font>,3,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 2
<b>by </b><i><a class="txt" href="idea_1.html#E1:39_1_1_2_1"><i><font color="Green" title="E37">A11</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
;<br/>
<b>hence </b><a NAME="E3:39_1_1_2_1"/>
 <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c3">i</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c3">i</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 2
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:39_1_1_2"/>
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 &amp; <font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 implies  <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c3">i</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c3">i</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c3">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_1_2_2"/><i><font color="Green" title="E37">A12</font></i>: 
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 &amp; <font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 )
 ;<br/>

<b>then </b> <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c3">i</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c3">i</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> = 
 <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c3">i</font>,3,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
<b>by </b><i><a class="txt" href="idea_1.html#E1:39_1_1_2_2"><i><font color="Green" title="E37">A12</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
;<br/>
<b>hence </b><a NAME="E3:39_1_1_2_2"/>
 <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c3">i</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c3">i</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:39_1_1_2"/><b>then </b>
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 &amp; <font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 implies  <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c3">i</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c3">i</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c3">i</font> )
 
<b>by </b><i><a class="txt" href="idea_1.html#E4:39_1_1"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="idea_1.html#E1:39_1_1_2"><i><font color="Green" title="E34">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:39_1_1_2"/>
 <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c3">i</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Maroon" title="c3">i</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E4:39_1_1"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="idea_1.html#E1:39_1_1_2"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="idea_1.html#E2:39_1_1_2"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="idea_1.html#E3:39_1_1_2"><i><font color="Green" title="E36">A10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:39_1_1"/><i><font color="Green" title="E34">A13</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> <font color="Maroon" title="c1">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">m</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c2">z</font>
;<br/>

<b>thus </b><a NAME="E7:39_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">z</font> <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="c1">m</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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> <font color="Maroon" title="c1">m</font> holds <br/><font color="Maroon" title="c2">z</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Olive" title="b1">i</font>,2,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <font color="Olive" title="b1">i</font>,3,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="idea_1.html#E3:39_1_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="idea_1.html#E5:39_1_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="idea_1.html#E6:39_1_1"><i><font color="Green" title="E34">A13</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>


</div>
