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

<b>let </b><font color="Maroon" title="c1">n5</font>, <font color="Maroon" title="c2">n4</font>, <font color="Maroon" title="c3">n3</font>, <font color="Maroon" title="c4">n2</font>, <font color="Maroon" title="c5">n1</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="E1:59"/><i><font color="Green" title="E36">A1</font></i>: 
 <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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">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">n5</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<a NAME="E1:59_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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c1">n5</font>
 
<b>by </b><i><a class="ref" href="genealg1.html#T32" target="_self" title="GENEALG1:th.32">Th32</a></i>;<br/>
<b>hence </b><a NAME="E2:59_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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">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">n5</font>
 <b>by </b><i><a class="ref" href="genealg1.html#T32" target="_self" title="GENEALG1:th.32">Th32</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:59"/><i><font color="Green" title="E37">A2</font></i>: 
 <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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c2">n4</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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">n4</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>proof </b></a><div class="add">

<a NAME="E1:59_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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c2">n4</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c2">n4</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">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c2">n4</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="genealg1.html#T31" target="_self" title="GENEALG1:th.31">Th31</a></i>;<br/>
<b>hence </b><a NAME="E2:59_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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c2">n4</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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">n4</font>
 <b>by </b><i><a class="ref" href="genealg1.html#T4" target="_self" title="GENEALG1:th.4">Th4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:59"/><i><font color="Green" title="E38">A3</font></i>: 
 <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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c3">n3</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="59_3"><b>proof </b></a><div class="add">

<a NAME="E1:59_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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c3">n3</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c3">n3</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="genealg1.html#T31" target="_self" title="GENEALG1:th.31">Th31</a></i>;<br/>
<b>hence </b><a NAME="E2:59_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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c3">n3</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="ref" href="genealg1.html#T4" target="_self" title="GENEALG1:th.4">Th4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:59"/><i><font color="Green" title="E39">A4</font></i>: 
 <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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c4">n2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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">n2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_4"><b>proof </b></a><div class="add">

<a NAME="E1:59_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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c4">n2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c4">n2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<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">n2</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="genealg1.html#T31" target="_self" title="GENEALG1:th.31">Th31</a></i>;<br/>
<b>hence </b><a NAME="E2:59_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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c4">n2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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">n2</font>
 <b>by </b><i><a class="ref" href="genealg1.html#T4" target="_self" title="GENEALG1:th.4">Th4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:59"/>
 <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="c5">n1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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">n1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_5"><b>proof </b></a><div class="add">

<a NAME="E1:59_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="c5">n1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c5">n1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<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="c5">n1</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="genealg1.html#T31" target="_self" title="GENEALG1:th.31">Th31</a></i>;<br/>
<b>hence </b><a NAME="E2:59_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="c5">n1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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">n1</font>
 <b>by </b><i><a class="ref" href="genealg1.html#T4" target="_self" title="GENEALG1:th.4">Th4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:59"/>
(  <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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">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">n5</font> &amp;  <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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c2">n4</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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">n4</font> &amp;  <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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c3">n3</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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;  <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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c4">n2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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">n2</font> &amp;  <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="c5">n1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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">n1</font> )
 <b>by </b><i><a class="txt" href="genealg1.html#E1:59"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="genealg1.html#E2:59"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="genealg1.html#E3:59"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="genealg1.html#E4:59"><i><font color="Green" title="E39">A4</font></i></a></i>;<br/>


</div>
