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<span class="kw">let </span><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>, <font color="Maroon" title="c5">n5</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K11" title="GENEALG1:func.11">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</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#K11" title="GENEALG1:func.11">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</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#K11" title="GENEALG1:func.11">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c4">n4</font>,<font color="Maroon" title="c5">n5</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#K11" title="GENEALG1:func.11">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c5">n5</font>) ) &amp; ( <font color="Maroon" title="c5">n5</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#K11" title="GENEALG1:func.11">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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>) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">S</font> be   <a href="genealg1.html#NM1" title="GENEALG1:NM.1">Gene-Set</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c6">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#K11" title="GENEALG1:func.11">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</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#K11" title="GENEALG1:func.11">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</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#K11" title="GENEALG1:func.11">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c4">n4</font>,<font color="Maroon" title="c5">n5</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#K11" title="GENEALG1:func.11">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c5">n5</font>) ) &amp; ( <font color="Maroon" title="c5">n5</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#K11" title="GENEALG1:func.11">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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>) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">p1</font>, <font color="Maroon" title="c8">p2</font> be    <a href="genealg1.html#M1" title="GENEALG1:mode.1">Individual</a> of <font color="Maroon" title="c6">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c7">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</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="c7">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</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="c7">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</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="c7">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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="c5">n5</font>) ) &amp; ( <font color="Maroon" title="c5">n5</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">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>) ) )</span><br/>















<a NAME="E1:61"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
( <font color="Maroon" title="c5">n5</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">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="61_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:61_1"/>
<font color="Maroon" title="c5">n5</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">p1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>)</span><br/>

<a NAME="E2:61_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">n5</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">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="genealg1.html#D1" target="_self" onmouseover="rs('genealg1/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E3:61_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A2</font></span>: 
<font color="Maroon" title="c5">n5</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#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="genealg1.html#D1" target="_self" onmouseover="rs('genealg1/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E4:61_1"/>
 <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</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#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>)</span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c7">p1</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>)</span>)</span>,<font color="Maroon" title="c5">n5</font>)
 
;<br/>
<span class="kw">hence </span><a NAME="E5:61_1"/>
 <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="genealg1.html#T5" target="_self" onmouseover="rs('genealg1/T5')" onmouseout="rh()">Th5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:61"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
( <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">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font>) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="61_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:61_2"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
<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">p1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font>)</span><br/>

<a NAME="E2:61_2"/><span class="kw">then </span>
<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">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="genealg1.html#D1" target="_self" onmouseover="rs('genealg1/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E3:61_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A5</font></span>: 
<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="c8">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="genealg1.html#D1" target="_self" onmouseover="rs('genealg1/D1')" onmouseout="rh()">Def1</a></span>;<br/>
 <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) = 
 <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="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c7">p1</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>)</span>)</span>,<font color="Maroon" title="c5">n5</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_2"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="genealg1.html#T33" target="_self" onmouseover="rs('genealg1/T33')" onmouseout="rh()">Th33</a></span>
<br/>.= 
 <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="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> (<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>)</span>)</span>,<font color="Maroon" title="c5">n5</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_2"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="ref" href="genealg1.html#T33" target="_self" onmouseover="rs('genealg1/T33')" onmouseout="rh()">Th33</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:61_2"/>
 <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:61"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
( <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="c7">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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="c5">n5</font>) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="61_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:61_3"/><span class="lab"><font color="Green" title="E40">A7</font></span>: 
<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="c7">p1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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="c5">n5</font>)</span><br/>

<a NAME="E2:61_3"/><span class="kw">then </span>
<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">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="genealg1.html#D1" target="_self" onmouseover="rs('genealg1/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E3:61_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A8</font></span>: 
<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="c8">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="genealg1.html#D1" target="_self" onmouseover="rs('genealg1/D1')" onmouseout="rh()">Def1</a></span>;<br/>
 <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) = 
 <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="c7">p1</font>,<font color="Maroon" title="c8">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#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c7">p1</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>)</span>)</span>,<font color="Maroon" title="c5">n5</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_3"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="ref" href="genealg1.html#T33" target="_self" onmouseover="rs('genealg1/T33')" onmouseout="rh()">Th33</a></span>
<br/>.= 
 <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="c7">p1</font>,<font color="Maroon" title="c8">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="c8">p2</font>,<font color="Maroon" title="c7">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="c5">n5</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_3"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="genealg1.html#T33" target="_self" onmouseover="rs('genealg1/T33')" onmouseout="rh()">Th33</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:61_3"/>
 <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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="c5">n5</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:61"/><span class="lab"><font color="Green" title="E40">A9</font></span>: 
( <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">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font>) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="61_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:61_4"/><span class="lab"><font color="Green" title="E41">A10</font></span>: 
<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">p1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font>)</span><br/>

<a NAME="E2:61_4"/><span class="kw">then </span>
<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">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="genealg1.html#D1" target="_self" onmouseover="rs('genealg1/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E3:61_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A11</font></span>: 
<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="c8">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="genealg1.html#D1" target="_self" onmouseover="rs('genealg1/D1')" onmouseout="rh()">Def1</a></span>;<br/>
 <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) = 
 <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="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c4">n4</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c7">p1</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>)</span>)</span>,<font color="Maroon" title="c5">n5</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_4"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="ref" href="genealg1.html#T33" target="_self" onmouseover="rs('genealg1/T33')" onmouseout="rh()">Th33</a></span>
<br/>.= 
 <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="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c4">n4</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> (<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c4">n4</font>)</span>)</span>,<font color="Maroon" title="c5">n5</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_4"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="ref" href="genealg1.html#T33" target="_self" onmouseover="rs('genealg1/T33')" onmouseout="rh()">Th33</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:61_4"/>
 <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:61"/>
( <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">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font>) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="61_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:61_5"/><span class="lab"><font color="Green" title="E41">A12</font></span>: 
<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">p1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font>)</span><br/>

<a NAME="E2:61_5"/><span class="kw">then </span>
<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">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="genealg1.html#D1" target="_self" onmouseover="rs('genealg1/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E3:61_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A13</font></span>: 
<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="c8">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="genealg1.html#D1" target="_self" onmouseover="rs('genealg1/D1')" onmouseout="rh()">Def1</a></span>;<br/>
 <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) = 
 <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="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c7">p1</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>)</span>)</span>,<font color="Maroon" title="c5">n5</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_5"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="ref" href="genealg1.html#T33" target="_self" onmouseover="rs('genealg1/T33')" onmouseout="rh()">Th33</a></span>
<br/>.= 
 <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="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K9" title="GENEALG1:func.9">crossover</a> (<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>)</span>)</span>,<font color="Maroon" title="c5">n5</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_5"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="ref" href="genealg1.html#T33" target="_self" onmouseover="rs('genealg1/T33')" onmouseout="rh()">Th33</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:61_5"/>
 <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:61"/>
( ( <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">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</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="c7">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c3">n3</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</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="c7">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c1">n1</font>,<font color="Maroon" title="c2">n2</font>,<font color="Maroon" title="c4">n4</font>,<font color="Maroon" title="c5">n5</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="c7">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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="c5">n5</font>) ) &amp; ( <font color="Maroon" title="c5">n5</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">p1</font> implies  <a href="genealg1.html#K11" title="GENEALG1:func.11">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>,<font color="Maroon" title="c5">n5</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="genealg1.html#K10" title="GENEALG1:func.10">crossover</a> (<font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">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>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E4:61"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E37">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
