<?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">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="c5">S</font> be   <a href="genealg1.html#NM1" title="GENEALG1:NM.1">Gene-Set</a>;<br/><b>let </b><font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c7">p2</font> be    <a href="genealg1.html#M1" title="GENEALG1:mode.1">Individual</a> of <font color="Maroon" title="c5">S</font>;<br/>













<b>thus </b><a NAME="E1:52"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K8" title="GENEALG1:func.8">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">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> = 
 <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="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c2">n3</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n3</font></span>)</span>,<font color="Maroon" title="c3">n4</font>
<b>by </b><i><a class="ref" href="genealg1.html#T27" target="_self" title="GENEALG1:th.27">Th27</a></i>
<br/>.= 
 <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="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c2">n3</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c2">n3</font></span>)</span>,<font color="Maroon" title="c3">n4</font>
<b>by </b><i><a class="ref" href="genealg1.html#T27" target="_self" title="GENEALG1:th.27">Th27</a></i>
;<br/>
<b>hence </b><a NAME="E2:52_1"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K8" title="GENEALG1:func.8">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c3">n4</font>
 ;<br/>


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

<b>thus </b><a NAME="E2:52"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K8" title="GENEALG1:func.8">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c3">n4</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>proof </b></a><div class="add">

 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n4</font> = 
 <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="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c4">n2</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c1">n1</font></span>)</span>,<font color="Maroon" title="c3">n4</font>
<b>by </b><i><a class="ref" href="genealg1.html#T27" target="_self" title="GENEALG1:th.27">Th27</a></i>
<br/>.= 
 <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="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c4">n2</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c4">n2</font></span>)</span>,<font color="Maroon" title="c3">n4</font>
<b>by </b><i><a class="ref" href="genealg1.html#T27" target="_self" title="GENEALG1:th.27">Th27</a></i>
;<br/>
<b>hence </b><a NAME="E2:52_2"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K8" title="GENEALG1:func.8">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c3">n4</font>
 ;<br/>


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

<b>thus </b><a NAME="E3:52"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c1">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K8" title="GENEALG1:func.8">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c2">n3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3"><b>proof </b></a><div class="add">

 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c1">n1</font> = 
 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c2">n3</font>
<b>by </b><i><a class="ref" href="genealg1.html#T37" target="_self" title="GENEALG1:th.37">Th37</a></i>
<br/>.= 
 <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="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c4">n2</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n2</font></span>)</span>,<font color="Maroon" title="c2">n3</font>
<b>by </b><i><a class="ref" href="genealg1.html#T27" target="_self" title="GENEALG1:th.27">Th27</a></i>
<br/>.= 
 <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="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c4">n2</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c4">n2</font></span>)</span>,<font color="Maroon" title="c2">n3</font>
<b>by </b><i><a class="ref" href="genealg1.html#T27" target="_self" title="GENEALG1:th.27">Th27</a></i>
;<br/>
<b>hence </b><a NAME="E2:52_3"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c1">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K8" title="GENEALG1:func.8">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c2">n3</font>
 ;<br/>


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

<b>thus </b><a NAME="E4:52"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c3">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K8" title="GENEALG1:func.8">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n4</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_4"><b>proof </b></a><div class="add">

 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c3">n4</font> = 
 <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="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c4">n2</font></span>)</span>,<font color="Maroon" title="c3">n4</font>
<b>by </b><i><a class="ref" href="genealg1.html#T27" target="_self" title="GENEALG1:th.27">Th27</a></i>
<br/>.= 
 <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="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">n1</font></span>)</span>,<font color="Maroon" title="c3">n4</font>
<b>by </b><i><a class="ref" href="genealg1.html#T27" target="_self" title="GENEALG1:th.27">Th27</a></i>
;<br/>
<b>hence </b><a NAME="E2:52_4"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c3">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K8" title="GENEALG1:func.8">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n4</font>
 ;<br/>


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

<b>thus </b><a NAME="E5:52"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c4">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K8" title="GENEALG1:func.8">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_5"><b>proof </b></a><div class="add">

 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c4">n2</font> = 
 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c2">n3</font>
<b>by </b><i><a class="ref" href="genealg1.html#T37" target="_self" title="GENEALG1:th.37">Th37</a></i>
<br/>.= 
 <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="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c4">n2</font></span>)</span>,<font color="Maroon" title="c2">n3</font>
<b>by </b><i><a class="ref" href="genealg1.html#T27" target="_self" title="GENEALG1:th.27">Th27</a></i>
<br/>.= 
 <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="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">n1</font></span>)</span>,<font color="Maroon" title="c2">n3</font>
<b>by </b><i><a class="ref" href="genealg1.html#T27" target="_self" title="GENEALG1:th.27">Th27</a></i>
;<br/>
<b>hence </b><a NAME="E2:52_5"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c4">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K8" title="GENEALG1:func.8">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n3</font>
 ;<br/>


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

 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c2">n3</font> = 
 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c4">n2</font>
<b>by </b><i><a class="ref" href="genealg1.html#T37" target="_self" title="GENEALG1:th.37">Th37</a></i>
<br/>.= 
 <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="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c2">n3</font></span>)</span>,<font color="Maroon" title="c4">n2</font>
<b>by </b><i><a class="ref" href="genealg1.html#T27" target="_self" title="GENEALG1:th.27">Th27</a></i>
<br/>.= 
 <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="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">n1</font></span>)</span>,<font color="Maroon" title="c4">n2</font>
<b>by </b><i><a class="ref" href="genealg1.html#T27" target="_self" title="GENEALG1:th.27">Th27</a></i>
;<br/>
<b>hence </b><a NAME="E7:52"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c2">n3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K8" title="GENEALG1:func.8">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c4">n2</font>
 ;<br/>


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