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

<b>let </b><font color="Maroon" title="c1">n1</font>, <font color="Maroon" title="c2">n2</font>, <font color="Maroon" title="c3">n3</font>, <font color="Maroon" title="c4">n4</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/> ( ( <font color="Maroon" title="c1">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p1</font> implies  <a href="genealg1.html#K10" title="GENEALG1:func.10">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="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> ) &amp; ( <font color="Maroon" title="c2">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p1</font> implies  <a href="genealg1.html#K10" title="GENEALG1:func.10">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="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">n3</font>,<font color="Maroon" title="c4">n4</font> ) &amp; ( <font color="Maroon" title="c3">n3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p1</font> implies  <a href="genealg1.html#K10" title="GENEALG1:func.10">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="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c2">n2</font>,<font color="Maroon" title="c4">n4</font> ) &amp; ( <font color="Maroon" title="c4">n4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p1</font> implies  <a href="genealg1.html#K10" title="GENEALG1:func.10">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="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c2">n2</font>,<font color="Maroon" title="c3">n3</font> ) )</span><br/><b>let </b><font color="Maroon" title="c5">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="c5">S</font> holds <br/> ( ( <font color="Maroon" title="c1">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p1</font> implies  <a href="genealg1.html#K10" title="GENEALG1:func.10">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="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> ) &amp; ( <font color="Maroon" title="c2">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p1</font> implies  <a href="genealg1.html#K10" title="GENEALG1:func.10">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="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">n3</font>,<font color="Maroon" title="c4">n4</font> ) &amp; ( <font color="Maroon" title="c3">n3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p1</font> implies  <a href="genealg1.html#K10" title="GENEALG1:func.10">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="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c2">n2</font>,<font color="Maroon" title="c4">n4</font> ) &amp; ( <font color="Maroon" title="c4">n4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p1</font> implies  <a href="genealg1.html#K10" title="GENEALG1:func.10">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="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c2">n2</font>,<font color="Maroon" title="c3">n3</font> ) )</span><br/><b>let </b><font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c7">p2</font> be    <a href="genealg1.html#M1" title="GENEALG1:mode.1">Individual</a> of <font color="Maroon" title="c5">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c1">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p1</font> implies  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> ) &amp; ( <font color="Maroon" title="c2">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p1</font> implies  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> ) &amp; ( <font color="Maroon" title="c3">n3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p1</font> implies  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c4">n4</font> ) &amp; ( <font color="Maroon" title="c4">n4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p1</font> implies  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font> ) )</span><br/>













<a NAME="E1:47"/><i><font color="Green" title="E29">A1</font></i>: 
( <font color="Maroon" title="c1">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p1</font> implies  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_1"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c1">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font></span><br/>

<a NAME="E2:47_1"/><b>then </b>
<font color="Maroon" title="c1">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">S</font>
 
<b>by </b><i><a class="ref" href="genealg1.html#D1" target="_self" title="GENEALG1:def.1">Def1</a></i>;<br/>
<a NAME="E3:47_1"/><b>then </b><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c1">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p2</font>
 
<b>by </b><i><a class="ref" href="genealg1.html#D1" target="_self" title="GENEALG1:def.1">Def1</a></i>;<br/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> = 
 <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K8" title="GENEALG1:func.8">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font></span>)</span>,<font color="Maroon" title="c4">n4</font>
<b>by </b><i><a class="txt" href="#E1:47_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="genealg1.html#T18" target="_self" title="GENEALG1:th.18">Th18</a></i>
<br/>.= 
 <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K8" title="GENEALG1:func.8">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K8" title="GENEALG1:func.8">crossover</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font></span>)</span>,<font color="Maroon" title="c4">n4</font>
<b>by </b><i><a class="txt" href="#E3:47_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="genealg1.html#T18" target="_self" title="GENEALG1:th.18">Th18</a></i>
;<br/>
<b>hence </b><a NAME="E5:47_1"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>
 ; <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>
<a NAME="E2:47"/><i><font color="Green" title="E30">A4</font></i>: 
( <font color="Maroon" title="c2">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p1</font> implies  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_2"/><i><font color="Green" title="E31">A5</font></i>: 
<font color="Maroon" title="c2">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font></span><br/>

<a NAME="E2:47_2"/><b>then </b>
<font color="Maroon" title="c2">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">S</font>
 
<b>by </b><i><a class="ref" href="genealg1.html#D1" target="_self" title="GENEALG1:def.1">Def1</a></i>;<br/>
<a NAME="E3:47_2"/><b>then </b><i><font color="Green" title="E32">A6</font></i>: 
<font color="Maroon" title="c2">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p2</font>
 
<b>by </b><i><a class="ref" href="genealg1.html#D1" target="_self" title="GENEALG1:def.1">Def1</a></i>;<br/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> = 
 <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K8" title="GENEALG1:func.8">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n3</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font></span>)</span>,<font color="Maroon" title="c4">n4</font>
<b>by </b><i><a class="txt" href="#E1:47_2"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="genealg1.html#T19" target="_self" title="GENEALG1:th.19">Th19</a></i>
<br/>.= 
 <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K8" title="GENEALG1:func.8">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n3</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K8" title="GENEALG1:func.8">crossover</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n3</font></span>)</span>,<font color="Maroon" title="c4">n4</font>
<b>by </b><i><a class="txt" href="#E3:47_2"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="genealg1.html#T19" target="_self" title="GENEALG1:th.19">Th19</a></i>
;<br/>
<b>hence </b><a NAME="E5:47_2"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>
 ; <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>
<a NAME="E3:47"/><i><font color="Green" title="E31">A7</font></i>: 
( <font color="Maroon" title="c3">n3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p1</font> implies  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c4">n4</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_3"/><i><font color="Green" title="E32">A8</font></i>: 
<font color="Maroon" title="c3">n3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c4">n4</font></span><br/>

<a NAME="E2:47_3"/><b>then </b>
<font color="Maroon" title="c3">n3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">S</font>
 
<b>by </b><i><a class="ref" href="genealg1.html#D1" target="_self" title="GENEALG1:def.1">Def1</a></i>;<br/>
<a NAME="E3:47_3"/><b>then </b><i><font color="Green" title="E33">A9</font></i>: 
<font color="Maroon" title="c3">n3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p2</font>
 
<b>by </b><i><a class="ref" href="genealg1.html#D1" target="_self" title="GENEALG1:def.1">Def1</a></i>;<br/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> = 
 <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K8" title="GENEALG1:func.8">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font></span>)</span>,<font color="Maroon" title="c4">n4</font>
<b>by </b><i><a class="txt" href="#E1:47_3"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="genealg1.html#T20" target="_self" title="GENEALG1:th.20">Th20</a></i>
<br/>.= 
 <a href="genealg1.html#K7" title="GENEALG1:func.7">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K8" title="GENEALG1:func.8">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K8" title="GENEALG1:func.8">crossover</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font></span>)</span>,<font color="Maroon" title="c4">n4</font>
<b>by </b><i><a class="txt" href="#E3:47_3"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="genealg1.html#T20" target="_self" title="GENEALG1:th.20">Th20</a></i>
;<br/>
<b>hence </b><a NAME="E5:47_3"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c4">n4</font>
 ; <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>
<a NAME="E4:47"/>
( <font color="Maroon" title="c4">n4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p1</font> implies  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_4"/>
<font color="Maroon" title="c4">n4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font></span><br/>

<a NAME="E2:47_4"/><b>then </b>
<font color="Maroon" title="c4">n4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">S</font>
 
<b>by </b><i><a class="ref" href="genealg1.html#D1" target="_self" title="GENEALG1:def.1">Def1</a></i>;<br/>
<a NAME="E3:47_4"/><b>then </b><i><font color="Green" title="E32">A10</font></i>: 
<font color="Maroon" title="c4">n4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="genealg1.html#D1" target="_self" title="GENEALG1:def.1">Def1</a></i>;<br/>
<a NAME="E4:47_4"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <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#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font></span>)</span>,<font color="Maroon" title="c4">n4</font>
 
;<br/>
<b>hence </b><a NAME="E5:47_4"/>
 <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>
 <b>by </b><i><a class="txt" href="#E3:47_4"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="genealg1.html#T5" target="_self" title="GENEALG1:th.5">Th5</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>hence </b><a NAME="E5:47"/>
( ( <font color="Maroon" title="c1">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p1</font> implies  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> ) &amp; ( <font color="Maroon" title="c2">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p1</font> implies  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> ) &amp; ( <font color="Maroon" title="c3">n3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p1</font> implies  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c4">n4</font> ) &amp; ( <font color="Maroon" title="c4">n4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p1</font> implies  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font> ) )
 <b>by </b><i><a class="txt" href="#E1:47"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E2:47"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="#E3:47"><i><font color="Green" title="E31">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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