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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">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">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="E1:25"/><i><font color="Green">E33</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:25"/><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:25"/>
 <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">n2</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">n2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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">n2</font></span>)</span>
 
;<br/>
<a NAME="E4:25"/><b>then </b><i><font color="Green">E35</font></i>: 
 <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">n2</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">n2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:25"/>
 <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">n2</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">n2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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">n2</font></span>)</span>
 
;<br/>
<a NAME="E6:25"/><b>then </b><i><font color="Green">E36</font></i>: 
 <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">n2</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">n2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1"/>
( ( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n2</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) or ( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n2</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) or <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n2</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_1_1_1"/>
( <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> or <font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</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="25_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_1_1_1_1"/><i><font color="Green">E38</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/><div class="add"><a NAME="E2:25_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/><b>then </b><i><font color="Green">E39</font></i>: <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> = 
<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/><a NAME="E4:25_1_1_1_1_1_1"/>
<font color="Maroon">p2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/><b>then </b><i><font color="Green">E40</font></i>: <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="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span> = 
<font color="Maroon">p2</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">p2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
;<br/><a NAME="E6:25_1_1_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">n2</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">n2</font></span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</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">n2</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/><b>then </b> <a href="genealg1.html#K8" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n1</font> = 
<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">n2</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">n2</font></span>)</span></span>)</span> <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/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</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">n2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
;<br/><b>hence </b><a NAME="E8:25_1_1_1_1_1_1"/>
 <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">n2</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K8" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n1</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_1_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font>
 ;<br/><div class="add"><a NAME="E2:25_1_1_1_1_1_2"/><b>then </b>
<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>
 <b>by </b><i>, </i>;<br/><a NAME="E3:25_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <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>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E4:25_1_1_1_1_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font>
 <b>by </b><i>, , , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:25_1_1_1_1_1_2"/><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">n2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:25_1_1_1_1_1_2"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E9:25_1_1_1_1_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:25_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
 <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">n2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E11:25_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <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/><a NAME="E12:25_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n2</font>
 <b>by </b><i/>;<br/><a NAME="E13:25_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n2</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E14:25_1_1_1_1_1_2"/>
 <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>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">n2</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#K17">|</a> <font color="Maroon">n2</font>
<b>by </b><i>, <a class="ref" href="finseq_5.html#T25">FINSEQ_5:25</a></i>
<br/>.= 
<font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</font>
<b>by </b><i>, <a class="ref" href="finseq_5.html#T80">FINSEQ_5:80</a></i>
;<br/><b>then </b><i><font color="Green">E71</font></i>:  <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">n2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n2</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>
<b>by </b><i>, </i>
<br/>.= 
<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">n2</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n2</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">p1</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#T45">FINSEQ_1:45</a></i>
;<br/><i><font color="Green">E72</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">n2</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">n2</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">n2</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n2</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</font></span>)</span> <a href="finseq_1.html#K8">^</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="rfinseq.html#K1">/^</a> <font color="Maroon">n2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="jordan3.html#T21">JORDAN3:21</a></i>
;<br/><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">n2</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">n2</font></span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n2</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_5.html#T39">FINSEQ_5:39</a></i>
<br/>.= 
<font color="Maroon">p1</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_6.html#T87">FINSEQ_6:87</a></i>
<br/>.= 
<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_1.html#T80">FINSEQ_1:80</a></i>
;<br/><b>hence </b><a NAME="E19:25_1_1_1_1_1_2"/>
 <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">n2</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K8" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n1</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:25_1_1_1"/>
 <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">n2</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K8" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n1</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n2</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_2_1_1"/>
( <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> or <font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</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="25_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_2_1_1_1"/><i><font color="Green">E74</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/><div class="add"><a NAME="E2:25_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E75</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>, </i>;<br/><a NAME="E3:25_1_1_2_1_1_1"/><i><font color="Green">E76</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>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:25_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E77</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>, </i>;<br/><a NAME="E5:25_1_1_2_1_1_1"/>
<font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/><b>then </b><i><font color="Green">E78</font></i>: <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> = 
<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/><a NAME="E7:25_1_1_2_1_1_1"/>
<font color="Maroon">p2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/><b>then </b><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="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span> = 
<font color="Maroon">p2</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">p2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
;<br/><a NAME="E9:25_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<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">n2</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/><a NAME="E10:25_1_1_2_1_1_1"/>
<font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</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">n2</font></span>)</span> = 
<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/><a NAME="E12:25_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E80</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">n2</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">n2</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>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/><a NAME="E13:25_1_1_2_1_1_1"/>
<font color="Maroon">p2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n2</font></span>)</span> = 
<font color="Maroon">p2</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">p2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
;<br/><a NAME="E15:25_1_1_2_1_1_1"/><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">n2</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">n2</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="E16:25_1_1_2_1_1_1"/>
 <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">n2</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K8" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n1</font>
 <b>by </b><i>, , , , <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_2_1_1_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font>
 ;<br/><div class="add"><a NAME="E2:25_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
 <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/><a NAME="E3:25_1_1_2_1_1_2"/><i><font color="Green">E83</font></i>: 
 <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>consider </b><font color="Maroon">i</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:25_1_1_2_1_1_2"/><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <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></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font>
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E7:25_1_1_2_1_1_2"/><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <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></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E8:25_1_1_2_1_1_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <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">n2</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_2_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_2_1_1_2_1_1_1"/>
( <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> or <font color="Maroon">n2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</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="25_1_1_2_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_2_1_1_2_1_1_1_1"/>
<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/><div class="add"><b>hence </b><a NAME="E2:25_1_1_2_1_1_2_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> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_2_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_2_1_1_2_1_1_1_2"/>
<font color="Maroon">n2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_1_1_2_1_1_2_1_1_1_2"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <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">n2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:25_1_1_2_1_1_2_1"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <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">n2</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E9:25_1_1_2_1_1_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <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">n2</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2_1_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_2_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_2_1_1_2_2_1_1"/>
( <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> or <font color="Maroon">n2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</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="25_1_1_2_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_2_1_1_2_2_1_1_1"/>
<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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_1_1_2_1_1_2_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> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n2</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_2_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_2_1_1_2_2_1_1_2"/>
<font color="Maroon">n2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_1_1_2_1_1_2_2_1_1_2"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <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">n2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:25_1_1_2_1_1_2_2"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <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">n2</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E10:25_1_1_2_1_1_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E11:25_1_1_2_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">n2</font> <a href="hidden.html#R1">=</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="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>then </b><i><font color="Green">E89</font></i>:  <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">n2</font> = 
<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"><span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_5.html#T39">FINSEQ_5:39</a></i>
<br/>.= 
<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"><span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_6.html#T87">FINSEQ_6:87</a></i>
<br/>.= 
<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"><span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n2</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>
;<br/><i><font color="Green">E90</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">n2</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">n2</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">n2</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font>
<b>by </b><i>, <a class="ref" href="finseq_5.html#T25">FINSEQ_5:25</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_5.html#T80">FINSEQ_5:80</a></i>
;<br/><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">n2</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">n2</font></span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font> = 
<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">n2</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n2</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n2</font></span>)</span>
<b>by </b><i><a class="ref" href="jordan3.html#T21">JORDAN3:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n1</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n2</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/><b>hence </b><a NAME="E15:25_1_1_2_1_1_2"/>
 <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">n2</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K8" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n1</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:25_1_1_2"/>
 <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">n2</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K8" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n1</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_3"><b>suppose </b></a><a NAME="E1:25_1_1_3"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>then </b> <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">n2</font> = 
 <a href="genealg1.html#K7" target="_self">crossover</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">n1</font>
<b>by </b><i/>
<br/>.= 
 <a href="genealg1.html#K8" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,0,<font color="Maroon">n1</font>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E3:25_1_1_3"/>
 <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">n2</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K8" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:25"/>
 <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">n2</font> <a href="hidden.html#R1">=</a>  <a href="genealg1.html#K8" target="_self">crossover</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">n2</font>,<font color="Maroon">n1</font>
 ;<br/>


</div>
