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<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">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="E1:24"/><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1_1"/>
( <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> or <font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_1_1"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font>
 ;<br/><div class="add"><a NAME="E2:24_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><b>hence </b><a NAME="E3:24_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T26">FINSEQ_5:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font>
 ;<br/><div class="add"><a NAME="E2:24_1_1_1_2"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/><a NAME="E3:24_1_1_1_2"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font>
 <b>by </b><i/>;<br/><a NAME="E4:24_1_1_1_2"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="genealg1.html#K7" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:24_1_1_1_2"/><b>then </b>
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span></span>)</span>
 ;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>
<br/>.= 
<font color="Maroon">p1</font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>
<br/>.= 
<font color="Maroon">p1</font> <a href="finseq_1.html#K7">^</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, <a class="ref" href="finseq_5.html#T35">FINSEQ_5:35</a></i>
<br/>.= 
<font color="Maroon">p1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
<br/>.= 
<font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>
;<br/><b>hence </b><a NAME="E7:24_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:24_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:24"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_2_1_1"/>
( <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font> or <font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1_1_1"><b>suppose </b></a><a NAME="E1:24_2_1_1_1"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font>
 ;<br/><div class="add"><a NAME="E2:24_2_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><b>hence </b><a NAME="E3:24_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T40">FINSEQ_5:40</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1_1_2"><b>suppose </b></a><a NAME="E1:24_2_1_1_2"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font>
 ;<br/><div class="add"><a NAME="E2:24_2_1_1_2"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/><a NAME="E3:24_2_1_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font>
 <b>by </b><i/>;<br/><a NAME="E4:24_2_1_1_2"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="genealg1.html#K7" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n1</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:24_2_1_1_2"/><b>then </b>
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span></span>)</span>
 ;<br/><a NAME="E6:24_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_5.html#T35">FINSEQ_5:35</a></i>;<br/><b>hence </b><a NAME="E7:24_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="finseq_5.html#T35">FINSEQ_5:35</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:24_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:24"/>
 <a href="genealg1.html#K8" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n1</font>,<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font>
 <b>by </b><i>, <a class="ref" href="rfinseq.html#T21">RFINSEQ:21</a></i>;<br/>


</div>
