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<div class="add">

<b>let </b><font color="Maroon">n1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n3</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n4</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n5</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">S</font> be   <a href="genealg1.html#NM1">Gene-Set</a>;<br/><b>let </b><font color="Maroon">p1</font> be    <a href="genealg1.html#M1" target="_self">Individual</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">p2</font> be    <a href="genealg1.html#M1" target="_self">Individual</a> of <font color="Maroon">S</font>;<br/>















<a NAME="E1:66"/><i><font color="Green">E33</font></i>: 
 <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

 <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font> = 
 <a href="genealg1.html#K7" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font></span>)</span>,<font color="Maroon">n5</font>
<b>by </b><i><a class="ref" href="genealg1.html#T1" target="_self">Th1</a></i>
<br/>.= 
 <a href="genealg1.html#K7" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font></span>)</span>,<font color="Maroon">n5</font>
<b>by </b><i><a class="ref" href="genealg1.html#T1" target="_self">Th1</a></i>
;<br/>
<b>hence </b><a NAME="E2:66_1"/>
 <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:66"/><i><font color="Green">E34</font></i>: 
 <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>proof </b></a><div class="add">

 <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font> = 
 <a href="genealg1.html#K7" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n4</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font></span>)</span>,<font color="Maroon">n5</font>
<b>by </b><i><a class="ref" href="genealg1.html#T1" target="_self">Th1</a></i>
<br/>.= 
 <a href="genealg1.html#K7" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n4</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n3</font>,<font color="Maroon">n2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n4</font></span>)</span>,<font color="Maroon">n5</font>
<b>by </b><i><a class="ref" href="genealg1.html#T1" target="_self">Th1</a></i>
;<br/>
<b>hence </b><a NAME="E2:66_2"/>
 <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:66"/><i><font color="Green">E35</font></i>: 
 <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n1</font>,<font color="Maroon">n5</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_3"><b>proof </b></a><div class="add">

 <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font> = 
 <a href="genealg1.html#K7" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n1</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font></span>)</span>,<font color="Maroon">n5</font>
<b>by </b><i><a class="ref" href="genealg1.html#T1" target="_self">Th1</a></i>
<br/>.= 
 <a href="genealg1.html#K7" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n1</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n4</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n1</font></span>)</span>,<font color="Maroon">n5</font>
<b>by </b><i><a class="ref" href="genealg1.html#T1" target="_self">Th1</a></i>
;<br/>
<b>hence </b><a NAME="E2:66_3"/>
 <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n1</font>,<font color="Maroon">n5</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:66"/>
 <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n5</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">q1</font> =  <a href="genealg1.html#K9" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>;<br/>
<b>set </b><font color="Maroon">q2</font> =  <a href="genealg1.html#K9" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>;<br/>
<a NAME="E1:66_4"/><i><font color="Green">E36</font></i>: 
 <a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K7" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K9" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K9" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font></span>)</span>,<font color="Maroon">n5</font>
 
;<br/>
 <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font> = 
 <a href="genealg1.html#K7" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n1</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font></span>)</span>,<font color="Maroon">n5</font>
<b>by </b><i><a class="ref" href="genealg1.html#T1" target="_self">Th1</a></i>
<br/>.= 
 <a href="genealg1.html#K7" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n1</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n1</font></span>)</span>,<font color="Maroon">n5</font>
<b>by </b><i><a class="ref" href="genealg1.html#T1" target="_self">Th1</a></i>
<br/>.= 
 <a href="genealg1.html#K8" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K9" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K9" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font></span>)</span>,<font color="Maroon">n1</font>,<font color="Maroon">n5</font>

<br/>.= 
 <a href="genealg1.html#K8" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K9" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K9" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font></span>)</span>,<font color="Maroon">n5</font>,<font color="Maroon">n1</font>
<b>by </b><i/>
<br/>.= 
 <a href="genealg1.html#K7" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n5</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font></span>)</span>,<font color="Maroon">n1</font>
<b>by </b><i>, <a class="ref" href="genealg1.html#T1" target="_self">Th1</a></i>
<br/>.= 
 <a href="genealg1.html#K7" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n5</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n5</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font></span>)</span>,<font color="Maroon">n1</font>
<b>by </b><i><a class="ref" href="genealg1.html#T1" target="_self">Th1</a></i>
;<br/>
<b>hence </b><a NAME="E3:66_4"/>
 <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n5</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n1</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:66"/>
(  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font> &amp;  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font> &amp;  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n1</font>,<font color="Maroon">n5</font> &amp;  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n5</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n1</font> )
 <b>by </b><i>, , </i>;<br/>


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