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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">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="E1:49"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> implies  <a href="genealg1.html#K10" 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> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K7" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n4</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> )
 ;<br/>

<a NAME="E2:49_1"/><b>then </b>
 <a href="genealg1.html#K10" 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> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K8" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:49_1"/>
 <a href="genealg1.html#K10" 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> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K7" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n4</font>
 <b>by </b><i><a class="ref" href="genealg1.html#T1" target="_self">Th1</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:49"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> implies  <a href="genealg1.html#K10" 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> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K7" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n3</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> )
 ;<br/>

<b>then </b> <a href="genealg1.html#K10" 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> = 
 <a href="genealg1.html#K8" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>
<b>by </b><i/>
<br/>.= 
 <a href="genealg1.html#K8" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<font color="Maroon">n3</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E3:49_2"/>
 <a href="genealg1.html#K10" 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> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K7" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n3</font>
 <b>by </b><i><a class="txt" href="genealg1.html#E1:49"><i><font color="Green">E33</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:49"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> implies  <a href="genealg1.html#K10" 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> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K7" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_3"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> )
 ;<br/>

<b>then </b> <a href="genealg1.html#K10" 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> = 
 <a href="genealg1.html#K8" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n4</font>
<b>by </b><i/>
<br/>.= 
 <a href="genealg1.html#K8" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<font color="Maroon">n2</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E3:49_3"/>
 <a href="genealg1.html#K10" 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> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K7" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:49"/>
( <font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> implies  <a href="genealg1.html#K10" 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> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K7" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_4"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> )
 ;<br/>

<b>then </b> <a href="genealg1.html#K10" 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> = 
 <a href="genealg1.html#K8" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n4</font>
<b>by </b><i/>
<br/>.= 
 <a href="genealg1.html#K8" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n4</font>,<font color="Maroon">n1</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E3:49_4"/>
 <a href="genealg1.html#K10" 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> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K7" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:49"/>
( ( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> implies  <a href="genealg1.html#K10" 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> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K7" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n4</font> ) &amp; ( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> implies  <a href="genealg1.html#K10" 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> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K7" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n3</font> ) &amp; ( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> implies  <a href="genealg1.html#K10" 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> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K7" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font> ) &amp; ( <font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">n4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> implies  <a href="genealg1.html#K10" 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> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K7" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font> ) )
 <b>by </b><i>, <a class="ref" href="genealg1.html#T1" target="_self">Th1</a>, <a class="txt" href="genealg1.html#E1:49"><i><font color="Green">E33</font></i></a></i>;<br/>


</div>
