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

<b>let </b><font color="Maroon" title="c1">n1</font>, <font color="Maroon" title="c2">n3</font>, <font color="Maroon" title="c3">n4</font>, <font color="Maroon" title="c4">n5</font>, <font color="Maroon" title="c5">n6</font>, <font color="Maroon" title="c6">n2</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="c7">S</font> be   <a href="genealg1.html#NM1" title="GENEALG1:NM.1">Gene-Set</a>;<br/><b>let </b><font color="Maroon" title="c8">p1</font>, <font color="Maroon" title="c9">p2</font> be    <a href="genealg1.html#M1" title="GENEALG1:mode.1">Individual</a> of <font color="Maroon" title="c7">S</font>;<br/>

















<b>thus </b><a NAME="E1:73"/>
 <a href="genealg1.html#K12" title="GENEALG1:func.12">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c4">n5</font>,<font color="Maroon" title="c5">n6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c4">n5</font>,<font color="Maroon" title="c5">n6</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </b></a><div class="add">

 <a href="genealg1.html#K12" title="GENEALG1:func.12">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c4">n5</font>,<font color="Maroon" title="c5">n6</font> = 
 <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c4">n5</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c4">n5</font></span>)</span>,<font color="Maroon" title="c5">n6</font>
<b>by </b><i><a class="ref" href="genealg1.html#T52" target="_self" title="GENEALG1:th.52">Th52</a></i>
<br/>.= 
 <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c4">n5</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c4">n5</font></span>)</span>,<font color="Maroon" title="c5">n6</font>
<b>by </b><i><a class="ref" href="genealg1.html#T52" target="_self" title="GENEALG1:th.52">Th52</a></i>
;<br/>
<b>hence </b><a NAME="E2:73_1"/>
 <a href="genealg1.html#K12" title="GENEALG1:func.12">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c4">n5</font>,<font color="Maroon" title="c5">n6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c4">n5</font>,<font color="Maroon" title="c5">n6</font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:73"/>
 <a href="genealg1.html#K12" title="GENEALG1:func.12">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c4">n5</font>,<font color="Maroon" title="c5">n6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c4">n5</font>,<font color="Maroon" title="c5">n6</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2"><b>proof </b></a><div class="add">

 <a href="genealg1.html#K12" title="GENEALG1:func.12">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c4">n5</font>,<font color="Maroon" title="c5">n6</font> = 
 <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c4">n5</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c4">n5</font></span>)</span>,<font color="Maroon" title="c5">n6</font>
<b>by </b><i><a class="ref" href="genealg1.html#T52" target="_self" title="GENEALG1:th.52">Th52</a></i>
<br/>.= 
 <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c4">n5</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c4">n5</font></span>)</span>,<font color="Maroon" title="c5">n6</font>
<b>by </b><i><a class="ref" href="genealg1.html#T52" target="_self" title="GENEALG1:th.52">Th52</a></i>
;<br/>
<b>hence </b><a NAME="E2:73_2"/>
 <a href="genealg1.html#K12" title="GENEALG1:func.12">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c4">n5</font>,<font color="Maroon" title="c5">n6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c4">n5</font>,<font color="Maroon" title="c5">n6</font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:73"/>
 <a href="genealg1.html#K12" title="GENEALG1:func.12">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n5</font>,<font color="Maroon" title="c5">n6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c4">n5</font>,<font color="Maroon" title="c5">n6</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_3"><b>proof </b></a><div class="add">

 <a href="genealg1.html#K12" title="GENEALG1:func.12">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n5</font>,<font color="Maroon" title="c5">n6</font> = 
 <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c4">n5</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n5</font></span>)</span>,<font color="Maroon" title="c5">n6</font>
<b>by </b><i><a class="ref" href="genealg1.html#T52" target="_self" title="GENEALG1:th.52">Th52</a></i>
<br/>.= 
 <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c4">n5</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c4">n5</font></span>)</span>,<font color="Maroon" title="c5">n6</font>
<b>by </b><i><a class="ref" href="genealg1.html#T52" target="_self" title="GENEALG1:th.52">Th52</a></i>
;<br/>
<b>hence </b><a NAME="E2:73_3"/>
 <a href="genealg1.html#K12" title="GENEALG1:func.12">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n5</font>,<font color="Maroon" title="c5">n6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c4">n5</font>,<font color="Maroon" title="c5">n6</font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:73"/>
 <a href="genealg1.html#K12" title="GENEALG1:func.12">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c5">n6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c5">n6</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_4"><b>proof </b></a><div class="add">

 <a href="genealg1.html#K12" title="GENEALG1:func.12">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c5">n6</font> = 
 <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c1">n1</font></span>)</span>,<font color="Maroon" title="c5">n6</font>
<b>by </b><i><a class="ref" href="genealg1.html#T52" target="_self" title="GENEALG1:th.52">Th52</a></i>
<br/>.= 
 <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font></span>)</span>,<font color="Maroon" title="c5">n6</font>
<b>by </b><i><a class="ref" href="genealg1.html#T52" target="_self" title="GENEALG1:th.52">Th52</a></i>
;<br/>
<b>hence </b><a NAME="E2:73_4"/>
 <a href="genealg1.html#K12" title="GENEALG1:func.12">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c5">n6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c5">n6</font>
 ;<br/>


</div><b>end;</b></div>

 <a href="genealg1.html#K12" title="GENEALG1:func.12">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c4">n5</font>,<font color="Maroon" title="c1">n1</font> = 
 <a href="genealg1.html#K12" title="GENEALG1:func.12">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c4">n5</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c1">n1</font>
<b>by </b><i><a class="ref" href="genealg1.html#T56" target="_self" title="GENEALG1:th.56">Th56</a></i>
<br/>.= 
 <a href="genealg1.html#K12" title="GENEALG1:func.12">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n5</font>
<b>by </b><i><a class="ref" href="genealg1.html#T56" target="_self" title="GENEALG1:th.56">Th56</a></i>
<br/>.= 
 <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c1">n1</font></span>)</span>,<font color="Maroon" title="c4">n5</font>
<b>by </b><i><a class="ref" href="genealg1.html#T52" target="_self" title="GENEALG1:th.52">Th52</a></i>
<br/>.= 
 <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font></span>)</span>,<font color="Maroon" title="c4">n5</font>
<b>by </b><i><a class="ref" href="genealg1.html#T52" target="_self" title="GENEALG1:th.52">Th52</a></i>
;<br/>
<b>hence </b><a NAME="E6:73"/>
 <a href="genealg1.html#K12" title="GENEALG1:func.12">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c4">n5</font>,<font color="Maroon" title="c1">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c6">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font>,<font color="Maroon" title="c4">n5</font>
 ;<br/>


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