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<b>let </b><font color="Maroon" title="c1">n1</font>, <font color="Maroon" title="c2">n3</font>, <font color="Maroon" title="c3">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="genealg1.html#NM1" title="GENEALG1:NM.1">Gene-Set</a><br/> for <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</font> being   <a href="genealg1.html#M1" title="GENEALG1:mode.1">Individual</a> of <font color="Olive" title="b1">S</font> holds <br/> (  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font>,<font color="Maroon" title="c2">n3</font> &amp;  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n2</font>,<font color="Maroon" title="c1">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font>,<font color="Maroon" title="c3">n2</font> &amp;  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n2</font>,<font color="Maroon" title="c3">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font>,<font color="Maroon" title="c1">n1</font> )</span><br/><b>let </b><font color="Maroon" title="c4">S</font> be   <a href="genealg1.html#NM1" title="GENEALG1:NM.1">Gene-Set</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font> being   <a href="genealg1.html#M1" title="GENEALG1:mode.1">Individual</a> of <font color="Maroon" title="c4">S</font> holds <br/> (  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>,<font color="Maroon" title="c2">n3</font> &amp;  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n2</font>,<font color="Maroon" title="c1">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>,<font color="Maroon" title="c3">n2</font> &amp;  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n2</font>,<font color="Maroon" title="c3">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>,<font color="Maroon" title="c1">n1</font> )</span><br/><b>let </b><font color="Maroon" title="c5">p1</font>, <font color="Maroon" title="c6">p2</font> be    <a href="genealg1.html#M1" title="GENEALG1:mode.1">Individual</a> of <font color="Maroon" title="c4">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n3</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="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">n3</font> &amp;  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n2</font>,<font color="Maroon" title="c1">n1</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="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c3">n2</font> &amp;  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n2</font>,<font color="Maroon" title="c3">n2</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="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c1">n1</font> )</span><br/>











<b>thus </b><a NAME="E1:40"/>
 <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n3</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="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">n3</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n2</font>,<font color="Maroon" title="c1">n1</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="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c3">n2</font> &amp;  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n2</font>,<font color="Maroon" title="c3">n2</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="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c1">n1</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<a NAME="E1:40_1"/>
 <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n3</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="c5">p1</font>,<span class="p1">(<span class="default"><a href="genealg1.html#K8" title="GENEALG1:func.8">crossover</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c1">n1</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#T12" target="_self" title="GENEALG1:th.12">Th12</a></i>;<br/>
<b>hence </b><a NAME="E2:40_1"/>
 <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n3</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="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">n3</font>
 <b>by </b><i><a class="ref" href="genealg1.html#T12" target="_self" title="GENEALG1:th.12">Th12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E2:40"/>
 <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n2</font>,<font color="Maroon" title="c1">n1</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="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c3">n2</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n2</font>,<font color="Maroon" title="c3">n2</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="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c1">n1</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

 <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n2</font>,<font color="Maroon" title="c1">n1</font> = 
 <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n2</font>
<b>by </b><i><a class="ref" href="genealg1.html#T25" target="_self" title="GENEALG1:th.25">Th25</a></i>
<br/>.= 
 <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c5">p1</font>,<span class="p1">(<span class="default"><a href="genealg1.html#K8" title="GENEALG1:func.8">crossover</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c1">n1</font></span>)</span>,<font color="Maroon" title="c3">n2</font>
<b>by </b><i><a class="ref" href="genealg1.html#T12" target="_self" title="GENEALG1:th.12">Th12</a></i>
;<br/>
<b>hence </b><a NAME="E2:40_2"/>
 <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n2</font>,<font color="Maroon" title="c1">n1</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="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c3">n2</font>
 <b>by </b><i><a class="ref" href="genealg1.html#T12" target="_self" title="GENEALG1:th.12">Th12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b> <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n2</font>,<font color="Maroon" title="c3">n2</font> = 
 <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c3">n2</font>,<font color="Maroon" title="c3">n2</font>,<font color="Maroon" title="c1">n1</font>
<b>by </b><i><a class="ref" href="genealg1.html#T26" target="_self" title="GENEALG1:th.26">Th26</a></i>
<br/>.= 
 <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c5">p1</font>,<span class="p1">(<span class="default"><a href="genealg1.html#K8" title="GENEALG1:func.8">crossover</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c3">n2</font>,<font color="Maroon" title="c3">n2</font></span>)</span>,<font color="Maroon" title="c1">n1</font>
<b>by </b><i><a class="ref" href="genealg1.html#T12" target="_self" title="GENEALG1:th.12">Th12</a></i>
<br/>.= 
 <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c1">n1</font>
<b>by </b><i><a class="ref" href="genealg1.html#T12" target="_self" title="GENEALG1:th.12">Th12</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
