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<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">k1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1_1"/>
( <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k2</font> or <font color="Maroon">k1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1"><b>case </b></a><a NAME="E1:74_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k2</font>
 ;<br/><div class="add"><a NAME="E2:74_1_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k1</font>,<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_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="E1:74_1_1_1_1_1"/>
( <font color="Maroon">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k2</font> or <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:74_1_1_1_1_1_1"/>
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k2</font>
 ;<br/><div class="add"><a NAME="E2:74_1_1_1_1_1_1"/><b>then </b>
 <a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k2</font>,<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:74_1_1_1_1_1_1"/>
 <a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k1</font>,<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k2</font>,<font color="Maroon">k1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_6.html#T29">FINSEQ_6:29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:74_1_1_1_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font>
 ;<br/><div class="add"><a NAME="E2:74_1_1_1_1_1_2"/>
( <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><a NAME="E3:74_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_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="E1:74_1_1_1_1_1_2_1_1"/>
( <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> 0 or ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> ) or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k1</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:74_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:74_1_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k1</font>,<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="finseq_1.html#K17">|</a> 1
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1_1_2_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="E1:74_1_1_1_1_1_2_1_1_1_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1_1_2_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:74_1_1_1_1_1_2_1_1_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:74_1_1_1_1_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/><a NAME="E3:74_1_1_1_1_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="finseq_1.html#K17">|</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:74_1_1_1_1_1_2_1_1_1_1_1_1"/>
 <a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k1</font>,<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k2</font>,<font color="Maroon">k1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan3.html#D1" target="_self">Def1</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1_1_2_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:74_1_1_1_1_1_2_1_1_1_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:74_1_1_1_1_1_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">f1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/><a NAME="E3:74_1_1_1_1_1_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">f1</font> <a href="finseq_1.html#K17">|</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T23">FINSEQ_5:23</a></i>;<br/><b>hence </b><a NAME="E4:74_1_1_1_1_1_2_1_1_1_1_1_2"/>
 <a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k1</font>,<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k2</font>,<font color="Maroon">k1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan3.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_5.html#T63">FINSEQ_5:63</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:74_1_1_1_1_1_2_1_1_1"/>
 <a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k1</font>,<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k2</font>,<font color="Maroon">k1</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:74_1_1_1_1_1_2_1_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> )
 ;<br/><div class="add"><a NAME="E2:74_1_1_1_1_1_2_1_1_2"/><b>then </b>
 <a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k1</font>,<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k1</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:74_1_1_1_1_1_2_1_1_2"/>
 <a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k1</font>,<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k2</font>,<font color="Maroon">k1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_5.html#T63">FINSEQ_5:63</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1_1_2_1_1_3"><b>case </b></a><a NAME="E1:74_1_1_1_1_1_2_1_1_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k1</font>
 ;<br/><div class="add"><a NAME="E2:74_1_1_1_1_1_2_1_1_3"/><b>then </b>
 <a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k1</font>,<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:74_1_1_1_1_1_2_1_1_3"/>
 <a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k1</font>,<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k2</font>,<font color="Maroon">k1</font></span>)</span>
 <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="E5:74_1_1_1_1_1_2"/>
 <a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k1</font>,<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k2</font>,<font color="Maroon">k1</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:74_1_1_1"/>
 <a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k1</font>,<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k2</font>,<font color="Maroon">k1</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_2"><b>case </b></a><a NAME="E1:74_1_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k2</font>
 ;<br/><div class="add"><a NAME="E2:74_1_1_2"/><b>then </b>
 <a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k1</font>,<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:74_1_1_2"/>
 <a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k1</font>,<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k2</font>,<font color="Maroon">k1</font></span>)</span>
 <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="E2:74"/>
 <a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k1</font>,<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f1</font>,<font color="Maroon">k2</font>,<font color="Maroon">k1</font></span>)</span>
 ;<br/>


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