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

<b>let </b><font color="Maroon">n1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n3</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n4</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n5</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n6</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">S</font> be   <a href="genealg1.html#NM1">Gene-Set</a>;<br/><b>let </b><font color="Maroon">p1</font> be    <a href="genealg1.html#M1" target="_self">Individual</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">p2</font> be    <a href="genealg1.html#M1" target="_self">Individual</a> of <font color="Maroon">S</font>;<br/>

















<a NAME="E1:71"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> implies  <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font>
 ;<br/>

<a NAME="E2:71_1"/><b>then </b>
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:71_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font>
 
<b>by </b><i/>;<br/>
 <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> = 
 <a href="genealg1.html#K7" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font></span>)</span>,<font color="Maroon">n6</font>
<b>by </b><i>, <a class="txt" href="genealg1.html#E2:71"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
 <a href="genealg1.html#K7" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font></span>)</span>,<font color="Maroon">n6</font>
<b>by </b><i><a class="txt" href="genealg1.html#E4:71"><i><font color="Green">E42</font></i></a>, <a class="txt" href="genealg1.html#E2:71"><i><font color="Green">E36</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:71_1"/>
 <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:71"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> implies  <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font>
 ;<br/>

<a NAME="E2:71_2"/><b>then </b>
<font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:71_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font>
 
<b>by </b><i/>;<br/>
 <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> = 
 <a href="genealg1.html#K7" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font></span>)</span>,<font color="Maroon">n6</font>
<b>by </b><i><a class="txt" href="genealg1.html#E4:71"><i><font color="Green">E42</font></i></a>, <a class="txt" href="genealg1.html#E2:71"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
 <a href="genealg1.html#K7" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n1</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font></span>)</span>,<font color="Maroon">n6</font>
<b>by </b><i>, <a class="txt" href="genealg1.html#E2:71"><i><font color="Green">E36</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:71_2"/>
 <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:71"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">n3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> implies  <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">n3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font>
 ;<br/>

<a NAME="E2:71_3"/><b>then </b>
<font color="Maroon">n3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:71_3"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">n3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font>
 
<b>by </b><i/>;<br/>
 <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> = 
 <a href="genealg1.html#K7" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font></span>)</span>,<font color="Maroon">n6</font>
<b>by </b><i>, <a class="txt" href="genealg1.html#E2:71"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
 <a href="genealg1.html#K7" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font></span>)</span>,<font color="Maroon">n6</font>
<b>by </b><i>, <a class="txt" href="genealg1.html#E2:71"><i><font color="Green">E36</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:71_3"/>
 <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:71"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">n4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> implies  <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_4"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">n4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font>
 ;<br/>

<a NAME="E2:71_4"/><b>then </b>
<font color="Maroon">n4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:71_4"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">n4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font>
 
<b>by </b><i/>;<br/>
 <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> = 
 <a href="genealg1.html#K7" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n5</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font></span>)</span>,<font color="Maroon">n6</font>
<b>by </b><i><a class="txt" href="genealg1.html#E3:71_4"><i><font color="Green">E44</font></i></a>, <a class="txt" href="genealg1.html#E2:71"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
 <a href="genealg1.html#K7" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n5</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n5</font></span>)</span>,<font color="Maroon">n6</font>
<b>by </b><i><a class="txt" href="genealg1.html#E5:71"><i><font color="Green">E45</font></i></a>, <a class="txt" href="genealg1.html#E2:71"><i><font color="Green">E36</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:71_4"/>
 <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:71"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">n5</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> implies  <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n6</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_5"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">n5</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font>
 ;<br/>

<a NAME="E2:71_5"/><b>then </b>
<font color="Maroon">n5</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:71_5"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">n5</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font>
 
<b>by </b><i/>;<br/>
 <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> = 
 <a href="genealg1.html#K7" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font></span>)</span>,<font color="Maroon">n6</font>
<b>by </b><i><a class="txt" href="genealg1.html#E5:71"><i><font color="Green">E45</font></i></a>, <a class="txt" href="genealg1.html#E2:71"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
 <a href="genealg1.html#K7" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K10" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font></span>)</span>,<font color="Maroon">n6</font>
<b>by </b><i><a class="txt" href="genealg1.html#E1:71_5"><i><font color="Green">E46</font></i></a>, <a class="txt" href="genealg1.html#E2:71"><i><font color="Green">E36</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:71_5"/>
 <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n6</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:71"/>
( <font color="Maroon">n6</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> implies  <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_6"/>
<font color="Maroon">n6</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font>
 ;<br/>

<a NAME="E2:71_6"/><b>then </b>
<font color="Maroon">n6</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:71_6"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">n6</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:71_6"/>
 <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K7" target="_self">crossover</a> <span class="p1">(<span class="default"><a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font></span>)</span>,<span class="p1">(<span class="default"><a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font></span>)</span>,<font color="Maroon">n6</font>
 
;<br/>
<b>hence </b><a NAME="E5:71_6"/>
 <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>
 <b>by </b><i><a class="txt" href="genealg1.html#E5:71"><i><font color="Green">E45</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:71"/>
( ( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> implies  <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> ) &amp; ( <font color="Maroon">n2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> implies  <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> ) &amp; ( <font color="Maroon">n3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> implies  <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> ) &amp; ( <font color="Maroon">n4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> implies  <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> ) &amp; ( <font color="Maroon">n5</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> implies  <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n6</font> ) &amp; ( <font color="Maroon">n6</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> implies  <a href="genealg1.html#K12" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font>,<font color="Maroon">n6</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K11" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font>,<font color="Maroon">n5</font> ) )
 <b>by </b><i>, , <a class="txt" href="genealg1.html#E4:71"><i><font color="Green">E42</font></i></a>, , </i>;<br/>


</div>
