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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> ;<br/><b>let </b><font color="Maroon" title="c4">S</font> be   <a href="genealg1.html#NM1" title="GENEALG1:NM.1">Gene-Set</a>;<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>;<br/>











<b>thus </b><a NAME="E1:53"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">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>,<font color="Maroon" title="c2">n3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<a NAME="E1:53_1"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">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>,<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="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">n3</font>,<font color="Maroon" title="c2">n3</font>
 
<b>by </b><i><a class="ref" href="genealg1.html#T38" target="_self" title="GENEALG1:th.38">Th38</a></i>;<br/>
<b>hence </b><a NAME="E2:53_1"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">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>,<font color="Maroon" title="c2">n3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font>
 <b>by </b><i><a class="ref" href="genealg1.html#T12" target="_self" title="GENEALG1:th.12">Th12</a></i>;<br/>


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

<b>thus </b><a NAME="E2:53"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">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>,<font color="Maroon" title="c3">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>proof </b></a><div class="add">

<a NAME="E1:53_2"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">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>,<font color="Maroon" title="c3">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="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c3">n2</font>,<font color="Maroon" title="c3">n2</font>
 
<b>by </b><i><a class="ref" href="genealg1.html#T38" target="_self" title="GENEALG1:th.38">Th38</a></i>;<br/>
<b>hence </b><a NAME="E2:53_2"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">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>,<font color="Maroon" title="c3">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font>
 <b>by </b><i><a class="ref" href="genealg1.html#T12" target="_self" title="GENEALG1:th.12">Th12</a></i>;<br/>


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

<a NAME="E3:53"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">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>,<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="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c3">n2</font>,<font color="Maroon" title="c3">n2</font>
 
<b>by </b><i><a class="ref" href="genealg1.html#T38" target="_self" title="GENEALG1:th.38">Th38</a></i>;<br/>
<b>hence </b><a NAME="E4:53"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">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>,<font color="Maroon" title="c1">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font>
 <b>by </b><i><a class="ref" href="genealg1.html#T12" target="_self" title="GENEALG1:th.12">Th12</a></i>;<br/>


</div>
