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

<b>let </b><font color="Maroon" title="c1">n1</font>, <font color="Maroon" title="c2">n2</font>, <font color="Maroon" title="c3">n3</font>, <font color="Maroon" title="c4">n4</font>, <font color="Maroon" title="c5">n5</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>let </b><font color="Maroon" title="c6">S</font> be   <a href="genealg1.html#NM1" title="GENEALG1:NM.1">Gene-Set</a>;<br/><b>let </b><font color="Maroon" title="c7">p1</font>, <font color="Maroon" title="c8">p2</font> be    <a href="genealg1.html#M1" title="GENEALG1:mode.1">Individual</a> of <font color="Maroon" title="c6">S</font>;<br/>















<a NAME="E1:64"/><i><font color="Green" title="E37">A1</font></i>: 
( <font color="Maroon" title="c1">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c2">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c3">n3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c4">n4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c5">n5</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_1"/><i><font color="Green" title="E38">A2</font></i>: 
( <font color="Maroon" title="c1">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c2">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c3">n3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c4">n4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> )
 ;<br/>

<a NAME="E2:64_1"/><b>then </b>
( <font color="Maroon" title="c1">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">S</font> &amp; <font color="Maroon" title="c2">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">S</font> &amp; <font color="Maroon" title="c3">n3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">S</font> &amp; <font color="Maroon" title="c4">n4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">S</font> )
 
<b>by </b><i><a class="ref" href="genealg1.html#D1" target="_self" title="GENEALG1:def.1">Def1</a></i>;<br/>
<a NAME="E3:64_1"/><b>then </b><i><font color="Green" title="E39">A3</font></i>: 
( <font color="Maroon" title="c1">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p2</font> &amp; <font color="Maroon" title="c2">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p2</font> &amp; <font color="Maroon" title="c3">n3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p2</font> &amp; <font color="Maroon" title="c4">n4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p2</font> )
 
<b>by </b><i><a class="ref" href="genealg1.html#D1" target="_self" title="GENEALG1:def.1">Def1</a></i>;<br/>
<a NAME="E4:64_1"/>
 <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font></span>)</span>,<font color="Maroon" title="c5">n5</font>
 
<b>by </b><i><a class="txt" href="genealg1.html#E1:64_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="genealg1.html#T36" target="_self" title="GENEALG1:th.36">Th36</a></i>;<br/>
<b>hence </b><a NAME="E5:64_1"/>
 <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c5">n5</font>
 <b>by </b><i><a class="txt" href="genealg1.html#E3:64_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="genealg1.html#T36" target="_self" title="GENEALG1:th.36">Th36</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:64"/><i><font color="Green" title="E38">A4</font></i>: 
( <font color="Maroon" title="c1">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c2">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c3">n3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c5">n5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c4">n4</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_2"/><i><font color="Green" title="E39">A5</font></i>: 
( <font color="Maroon" title="c1">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c2">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c3">n3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c5">n5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> )
 ;<br/>

<a NAME="E2:64_2"/><b>then </b>
<font color="Maroon" title="c5">n5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">S</font>
 
<b>by </b><i><a class="ref" href="genealg1.html#D1" target="_self" title="GENEALG1:def.1">Def1</a></i>;<br/>
<a NAME="E3:64_2"/><b>then </b><i><font color="Green" title="E40">A6</font></i>: 
<font color="Maroon" title="c5">n5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c4">n4</font></span>)</span>
 
<b>by </b><i><a class="ref" href="genealg1.html#D1" target="_self" title="GENEALG1:def.1">Def1</a></i>;<br/>
<a NAME="E4:64_2"/>
 <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c4">n4</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font></span>)</span>,<font color="Maroon" title="c5">n5</font>
 
<b>by </b><i><a class="txt" href="genealg1.html#E1:64_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="genealg1.html#T35" target="_self" title="GENEALG1:th.35">Th35</a></i>;<br/>
<b>hence </b><a NAME="E5:64_2"/>
 <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c4">n4</font>
 <b>by </b><i><a class="txt" href="genealg1.html#E3:64_2"><i><font color="Green" title="E40">A6</font></i></a>, <a class="ref" href="genealg1.html#T5" target="_self" title="GENEALG1:th.5">Th5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:64"/><i><font color="Green" title="E39">A7</font></i>: 
( <font color="Maroon" title="c1">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c2">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c4">n4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c5">n5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c3">n3</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_3"/><i><font color="Green" title="E40">A8</font></i>: 
( <font color="Maroon" title="c1">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c2">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c4">n4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c5">n5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> )
 ;<br/>

<a NAME="E2:64_3"/><b>then </b>
<font color="Maroon" title="c5">n5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">S</font>
 
<b>by </b><i><a class="ref" href="genealg1.html#D1" target="_self" title="GENEALG1:def.1">Def1</a></i>;<br/>
<a NAME="E3:64_3"/><b>then </b><i><font color="Green" title="E41">A9</font></i>: 
<font color="Maroon" title="c5">n5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c3">n3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="genealg1.html#D1" target="_self" title="GENEALG1:def.1">Def1</a></i>;<br/>
<a NAME="E4:64_3"/>
 <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c3">n3</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font></span>)</span>,<font color="Maroon" title="c5">n5</font>
 
<b>by </b><i><a class="txt" href="genealg1.html#E1:64_3"><i><font color="Green" title="E40">A8</font></i></a>, <a class="ref" href="genealg1.html#T35" target="_self" title="GENEALG1:th.35">Th35</a></i>;<br/>
<b>hence </b><a NAME="E5:64_3"/>
 <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c3">n3</font>
 <b>by </b><i><a class="txt" href="genealg1.html#E3:64_3"><i><font color="Green" title="E41">A9</font></i></a>, <a class="ref" href="genealg1.html#T5" target="_self" title="GENEALG1:th.5">Th5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:64"/><i><font color="Green" title="E40">A10</font></i>: 
( <font color="Maroon" title="c1">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c3">n3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c4">n4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c5">n5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c2">n2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_4"/><i><font color="Green" title="E41">A11</font></i>: 
( <font color="Maroon" title="c1">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c3">n3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c4">n4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c5">n5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> )
 ;<br/>

<a NAME="E2:64_4"/><b>then </b>
<font color="Maroon" title="c5">n5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">S</font>
 
<b>by </b><i><a class="ref" href="genealg1.html#D1" target="_self" title="GENEALG1:def.1">Def1</a></i>;<br/>
<a NAME="E3:64_4"/><b>then </b><i><font color="Green" title="E42">A12</font></i>: 
<font color="Maroon" title="c5">n5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c2">n2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="genealg1.html#D1" target="_self" title="GENEALG1:def.1">Def1</a></i>;<br/>
<a NAME="E4:64_4"/>
 <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c2">n2</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font></span>)</span>,<font color="Maroon" title="c5">n5</font>
 
<b>by </b><i><a class="txt" href="genealg1.html#E1:64_4"><i><font color="Green" title="E41">A11</font></i></a>, <a class="ref" href="genealg1.html#T35" target="_self" title="GENEALG1:th.35">Th35</a></i>;<br/>
<b>hence </b><a NAME="E5:64_4"/>
 <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c2">n2</font>
 <b>by </b><i><a class="txt" href="genealg1.html#E3:64_4"><i><font color="Green" title="E42">A12</font></i></a>, <a class="ref" href="genealg1.html#T5" target="_self" title="GENEALG1:th.5">Th5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:64"/>
( <font color="Maroon" title="c2">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c3">n3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c4">n4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c5">n5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_5"/><i><font color="Green" title="E41">A13</font></i>: 
( <font color="Maroon" title="c2">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c3">n3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c4">n4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c5">n5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> )
 ;<br/>

<a NAME="E2:64_5"/><b>then </b>
<font color="Maroon" title="c5">n5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">S</font>
 
<b>by </b><i><a class="ref" href="genealg1.html#D1" target="_self" title="GENEALG1:def.1">Def1</a></i>;<br/>
<a NAME="E3:64_5"/><b>then </b><i><font color="Green" title="E42">A14</font></i>: 
<font color="Maroon" title="c5">n5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="genealg1.html#D1" target="_self" title="GENEALG1:def.1">Def1</a></i>;<br/>
<a NAME="E4:64_5"/>
 <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font></span>)</span>,<font color="Maroon" title="c5">n5</font>
 
<b>by </b><i><a class="txt" href="genealg1.html#E1:64_5"><i><font color="Green" title="E41">A13</font></i></a>, <a class="ref" href="genealg1.html#T35" target="_self" title="GENEALG1:th.35">Th35</a></i>;<br/>
<b>hence </b><a NAME="E5:64_5"/>
 <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>
 <b>by </b><i><a class="txt" href="genealg1.html#E3:64_5"><i><font color="Green" title="E42">A14</font></i></a>, <a class="ref" href="genealg1.html#T5" target="_self" title="GENEALG1:th.5">Th5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:64"/>
( ( <font color="Maroon" title="c1">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c2">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c3">n3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c4">n4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c5">n5</font> ) &amp; ( <font color="Maroon" title="c1">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c2">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c3">n3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c5">n5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c4">n4</font> ) &amp; ( <font color="Maroon" title="c1">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c2">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c4">n4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c5">n5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c3">n3</font> ) &amp; ( <font color="Maroon" title="c1">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c3">n3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c4">n4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c5">n5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c2">n2</font> ) &amp; ( <font color="Maroon" title="c2">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c3">n3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c4">n4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c5">n5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font> ) )
 <b>by </b><i><a class="txt" href="genealg1.html#E1:64"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="genealg1.html#E2:64"><i><font color="Green" title="E38">A4</font></i></a>, <a class="txt" href="genealg1.html#E3:64"><i><font color="Green" title="E39">A7</font></i></a>, <a class="txt" href="genealg1.html#E4:64"><i><font color="Green" title="E40">A10</font></i></a></i>;<br/>


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