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<div class="add">

<b>let </b><font color="Maroon">f</font> be  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:31"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">g</font> <a href="jordan4.html#R2" target="_self">is_a_part&lt;_of</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i2</font> )
 ;<br/>

<a NAME="E2:31"/><b>then </b><i><font color="Green">E33</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Olive">i</font></span>)</span>,<font color="Maroon">f</font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="jordan4.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E3:31"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E4:31"/><b>then </b><i><font color="Green">E34</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:31"/><b>then </b><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/>
<a NAME="E6:31"/><i><font color="Green">E43</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a>, <a class="txt" href="jordan4.html#E5:31"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E7:31"/><i><font color="Green">E44</font></i>: 
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jordan4.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E8:31"/>
<font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E9:31"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E10:31"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a></i>;<br/>
<div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1"><b>case </b></a><a NAME="E1:31_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:31_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E3:31_1_1_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E4:31_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">f</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_2"><b>case </b></a><a NAME="E1:31_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:31_1_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i/>;<br/><a NAME="E3:31_1_1_2"/>
<font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><b>hence </b><a NAME="E4:31_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jordan4.html#T18" target="_self">Th18</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1"><b>case </b></a><a NAME="E1:31_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:31_2_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:31_2_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E4:31_2_1_1"/><b>then </b><i><font color="Green">E68</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:31_2_1_1"/><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T46">NAT_1:46</a>, <a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan4.html#E7:31"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E6:31_2_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="jordan4.html#T19" target="_self">Th19</a>, <a class="ref" href="jordan3.html#T14">JORDAN3:14</a></i>;<br/><b>hence </b><a NAME="E7:31_2_1_1"/>
 <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E6:31"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_2"><b>case </b></a><a NAME="E1:31_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:31_2_1_2"/><b>then </b><i><font color="Green">E79</font></i>: 
 <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i/>;<br/><a NAME="E3:31_2_1_2"/>
<font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E4:31_2_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:31_2_1_2"/>
 <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E6:31"><i><font color="Green">E43</font></i></a>, <a class="ref" href="jordan3.html#T14">JORDAN3:14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E13:31"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/>
<a NAME="E14:31"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:31"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="jordan3.html#T14">JORDAN3:14</a></i>;<br/>
<a NAME="E16:31"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="jordan4.html#E6:31"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_3_1"/>
( <font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">f</font> or <font color="Maroon">i2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">f</font> or <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a>  <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">f</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="31_3_1_1"><b>case </b></a><a NAME="E1:31_3_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:31_3_1_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jordan4.html#T18" target="_self">Th18</a>, <a class="ref" href="goboard7.html#T38">GOBOARD7:38</a></i>;<br/><a NAME="E3:31_3_1_1"/>
<font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:31_3_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E5:31_3_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jordan4.html#T19" target="_self">Th19</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E6:31_3_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jordan4.html#T18" target="_self">Th18</a>, , <a class="ref" href="jordan4.html#T20" target="_self">Th20</a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_2"><b>case </b></a><a NAME="E1:31_3_1_2"/>
<font color="Maroon">i2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:31_3_1_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan4.html#T14" target="_self">Th14</a>, <a class="txt" href="jordan4.html#E7:31"><i><font color="Green">E44</font></i></a>, <a class="ref" href="goboard7.html#T38">GOBOARD7:38</a></i>;<br/><a NAME="E3:31_3_1_2"/>
<font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:31_3_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><b>hence </b><a NAME="E5:31_3_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jordan4.html#T14" target="_self">Th14</a>, <a class="ref" href="jordan4.html#T18" target="_self">Th18</a>, <a class="ref" href="jordan4.html#T19" target="_self">Th19</a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_3"><b>case </b></a><a NAME="E1:31_3_1_3"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a>  <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">f</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_3_1_3_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_3_1_1_1"><b>case </b></a><a NAME="E1:31_3_1_3_1_1_1"/><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:31_3_1_3_1_1_1"/>
 ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 )
 <b>by </b><i><a class="ref" href="jordan4.html#T9" target="_self">Th9</a>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:31_3_1_3_1_1_1"/><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E5:31_3_1_3_1_1_1"/><i><font color="Green">E89</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="jordan4.html#T19" target="_self">Th19</a>, , <a class="ref" href="jordan4.html#T20" target="_self">Th20</a>, </i>;<br/><a NAME="E6:31_3_1_3_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E7:31_3_1_3_1_1_1"/><b>then </b><i><font color="Green">E90</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 <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="31_3_1_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_3_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="E1:31_3_1_3_1_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="jordan4.html#T22" target="_self">Th22</a>, <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="31_3_1_3_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:31_3_1_3_1_1_1_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:31_3_1_3_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan4.html#T19" target="_self">Th19</a>, , <a class="ref" href="jordan4.html#T20" target="_self">Th20</a>, </i>;<br/><a NAME="E3:31_3_1_3_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, </i>;<br/><a NAME="E4:31_3_1_3_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><b>hence </b><a NAME="E5:31_3_1_3_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan4.html#E2:31"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_3_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:31_3_1_3_1_1_1_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:31_3_1_3_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan4.html#T7" target="_self">Th7</a>, <a class="ref" href="jordan4.html#T19" target="_self">Th19</a>, , <a class="ref" href="jordan4.html#T20" target="_self">Th20</a>, </i>;<br/><a NAME="E3:31_3_1_3_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, </i>;<br/><a NAME="E4:31_3_1_3_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="jordan4.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E5:31_3_1_3_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E93</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E2:31"><i><font color="Green">E33</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E6:31_3_1_3_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E94</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 <b>by </b><i><a class="txt" href="jordan4.html#E2:31"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, , </i>;<br/><a NAME="E7:31_3_1_3_1_1_1_1_1_2"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3_1_3_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:31_3_1_3_1_1_1_1_1_2_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 ;<br/>

<a NAME="E2:31_3_1_3_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E95</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:31_3_1_3_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan4.html#E2:31"><i><font color="Green">E33</font></i></a>, <a class="ref" href="jordan4.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E4:31_3_1_3_1_1_1_1_1_2_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan4.html#E2:31"><i><font color="Green">E33</font></i></a>, <a class="ref" href="jordan4.html#T24" target="_self">Th24</a>, <a class="txt" href="jordan4.html#E10:31"><i><font color="Green">E54</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<i><font color="Green">E98</font></i>: <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="jordan4.html#E2:31"><i><font color="Green">E33</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> 1</span>)</span>

;<br/>
<a NAME="E6:31_3_1_3_1_1_1_1_1_2_1"/>
1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E7:31_3_1_3_1_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 0 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/>
<a NAME="E8:31_3_1_3_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font>
 
<b>by </b><i>, <a class="txt" href="jordan4.html#E12:31"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:31_3_1_3_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E100</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E10:31_3_1_3_1_1_1_1_1_2_1"/>
<font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E11:31_3_1_3_1_1_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E12:31_3_1_3_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3_1_3_1_1_1_1_1_2_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_3_1_1_1_1_1_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_3_1_3_1_1_1_1_1_2_1_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_3_1_1_1_1_1_2_1_2_1_1"><b>case </b></a><a NAME="E1:31_3_1_3_1_1_1_1_1_2_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="txt" href="jordan4.html#E5:31"><i><font color="Green">E35</font></i></a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>
;<br/><a NAME="E3:31_3_1_3_1_1_1_1_1_2_1_2_1_1"/><b>then </b>
<font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/><b>hence </b><a NAME="E4:31_3_1_3_1_1_1_1_1_2_1_2_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_3_1_1_1_1_1_2_1_2_1_2"><b>case </b></a><a NAME="E1:31_3_1_3_1_1_1_1_1_2_1_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 ;<br/><div class="add"><a NAME="E2:31_3_1_3_1_1_1_1_1_2_1_2_1_2"/><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3_1_3_1_1_1_1_1_2_1_2_1_2_1"><b>now </b></a><div class="add"><a NAME="E1:31_3_1_3_1_1_1_1_1_2_1_2_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a>, <a class="txt" href="jordan4.html#E10:31"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E2:31_3_1_3_1_1_1_1_1_2_1_2_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E3:31_3_1_3_1_1_1_1_1_2_1_2_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E4:31_3_1_3_1_1_1_1_1_2_1_2_1_2_1"/><b>then </b>
<font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E5:31_3_1_3_1_1_1_1_1_2_1_2_1_2_1"/><b>then </b>
<font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan4.html#T7" target="_self">Th7</a>, , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E6:31_3_1_3_1_1_1_1_1_2_1_2_1_2_1"/><b>then </b><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <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="31_3_1_3_1_1_1_1_1_2_1_2_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_3_1_1_1_1_1_2_1_2_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="E1:31_3_1_3_1_1_1_1_1_2_1_2_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 or <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 )
 </span><b>by </b><i><a class="ref" href="jordan4.html#T29" target="_self">Th29</a>, <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="31_3_1_3_1_1_1_1_1_2_1_2_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:31_3_1_3_1_1_1_1_1_2_1_2_1_2_1_1_1_1"/><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 ;<br/><div class="add"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan4.html#T7" target="_self">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="jordan4.html#T30" target="_self">Th30</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/><a NAME="E3:31_3_1_3_1_1_1_1_1_2_1_2_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:31_3_1_3_1_1_1_1_1_2_1_2_1_2_1_1_1_1"/>
 <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="jordan4.html#T30" target="_self">Th30</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_3_1_1_1_1_1_2_1_2_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:31_3_1_3_1_1_1_1_1_2_1_2_1_2_1_1_1_2"/><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 ;<br/><div class="add"><i><font color="Green">E106</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="jordan4.html#E5:31"><i><font color="Green">E35</font></i></a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="nat_1.html#T74">NAT_1:74</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="nat_1.html#T77">NAT_1:77</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="jordan4.html#T30" target="_self">Th30</a>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/><a NAME="E5:31_3_1_3_1_1_1_1_1_2_1_2_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E107</font></i>: 
 <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan4.html#T27" target="_self">Th27</a>, </i>;<br/><a NAME="E6:31_3_1_3_1_1_1_1_1_2_1_2_1_2_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
 ;<br/><a NAME="E7:31_3_1_3_1_1_1_1_1_2_1_2_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E108</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E5:31"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="jordan4.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E8:31_3_1_3_1_1_1_1_1_2_1_2_1_2_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T77">NAT_1:77</a></i>;<br/><b>then </b> <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="jordan4.html#T31" target="_self">Th31</a>, <a class="ref" href="nat_1.html#T74">NAT_1:74</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>

;<br/><b>hence </b><a NAME="E10:31_3_1_3_1_1_1_1_1_2_1_2_1_2_1_1_1_2"/>
 <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="jordan4.html#T30" target="_self">Th30</a>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:31_3_1_3_1_1_1_1_1_2_1_2_1_2_1"/>
 <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:31_3_1_3_1_1_1_1_1_2_1_2_1_2"/>
 <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E14:31_3_1_3_1_1_1_1_1_2_1"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E2:31"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, , <a class="ref" href="jordan4.html#T25" target="_self">Th25</a>, <a class="txt" href="jordan4.html#E10:31"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan4.html#E11:31"><i><font color="Green">E55</font></i></a>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:31_3_1_3_1_1_1_1_1_2"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font> )
 <b>by </b><i><a class="txt" href="jordan4.html#E2:31"><i><font color="Green">E33</font></i></a>, <a class="ref" href="jordan4.html#T24" target="_self">Th24</a>, <a class="txt" href="jordan4.html#E9:31"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_3_1_1_1_1_1_3"><b>case </b></a><a NAME="E1:31_3_1_3_1_1_1_1_1_3"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:31_3_1_3_1_1_1_1_1_3"/><b>then </b><i><font color="Green">E109</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:31_3_1_3_1_1_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E4:31_3_1_3_1_1_1_1_1_3"/><b>then </b><i><font color="Green">E110</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan4.html#T21" target="_self">Th21</a>, <a class="ref" href="jordan4.html#T23" target="_self">Th23</a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E5:31_3_1_3_1_1_1_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a>, <a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E6:31_3_1_3_1_1_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:31_3_1_3_1_1_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E8:31_3_1_3_1_1_1_1_1_3"/><b>then </b><i><font color="Green">E111</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E9:31_3_1_3_1_1_1_1_1_3"/><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E10:31_3_1_3_1_1_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><a NAME="E11:31_3_1_3_1_1_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E9:31"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E12:31_3_1_3_1_1_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E13:31_3_1_3_1_1_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan4.html#T25" target="_self">Th25</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E14:31_3_1_3_1_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jordan4.html#T7" target="_self">Th7</a>, <a class="ref" href="jordan4.html#T24" target="_self">Th24</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:31_3_1_3_1_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_3_1_1_2"><b>case </b></a><a NAME="E1:31_3_1_3_1_1_2"/><i><font color="Green">E113</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:31_3_1_3_1_1_2"/>
(  ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 ) or ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:31_3_1_3_1_1_2"/><i><font color="Green">E114</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jordan4.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E5:31_3_1_3_1_1_2"/><i><font color="Green">E115</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="jordan4.html#T19" target="_self">Th19</a>, , <a class="ref" href="jordan4.html#T20" target="_self">Th20</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3_1_3_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_3_1_3_1_1_2_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:31_3_1_3_1_1_2_1"/><b>then </b><i><font color="Green">E116</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font>
 <b>by </b><i>, <a class="ref" href="jordan4.html#T20" target="_self">Th20</a>, </i>;<br/><a NAME="E3:31_3_1_3_1_1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>hence </b><a NAME="E4:31_3_1_3_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jordan4.html#T22" target="_self">Th22</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><a NAME="E7:31_3_1_3_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E8:31_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><i><font color="Green">E118</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3_1_3_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_3_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_3_1_3_1_1_2_2_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="ref" href="jordan4.html#T22" target="_self">Th22</a>, <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="31_3_1_3_1_1_2_2_1_1"><b>case </b></a><a NAME="E1:31_3_1_3_1_1_2_2_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:31_3_1_3_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:31_3_1_3_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E4:31_3_1_3_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E5:31_3_1_3_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E6:31_3_1_3_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:31_3_1_3_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 ;<br/><a NAME="E8:31_3_1_3_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E120</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jordan4.html#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E9:31_3_1_3_1_1_2_2_1_1"/><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3_1_3_1_1_2_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_3_1_1_2_2_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="E1:31_3_1_3_1_1_2_2_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="jordan4.html#T24" target="_self">Th24</a>, <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="31_3_1_3_1_1_2_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:31_3_1_3_1_1_2_2_1_1_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_3_1_3_1_1_2_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan4.html#T8" target="_self">Th8</a>, , <a class="ref" href="jordan4.html#T20" target="_self">Th20</a>, <a class="ref" href="jordan4.html#T25" target="_self">Th25</a>, <a class="txt" href="jordan4.html#E9:31"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_3_1_1_2_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:31_3_1_3_1_1_2_2_1_1_1_1_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:31_3_1_3_1_1_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E123</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, , <a class="ref" href="jordan4.html#T20" target="_self">Th20</a>, </i>;<br/><a NAME="E3:31_3_1_3_1_1_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="jordan4.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E4:31_3_1_3_1_1_2_2_1_1_1_1_2"/>
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="jordan4.html#T7" target="_self">Th7</a>, <a class="txt" href="jordan4.html#E11:31"><i><font color="Green">E55</font></i></a></i>;<br/><a NAME="E5:31_3_1_3_1_1_2_2_1_1_1_1_2"/><b>then </b>
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E5:31"><i><font color="Green">E35</font></i></a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E6:31_3_1_3_1_1_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E126</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E7:31_3_1_3_1_1_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E127</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E8:31_3_1_3_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font>
 <b>by </b><i>, <a class="ref" href="jordan4.html#T20" target="_self">Th20</a>, <a class="ref" href="jordan4.html#T21" target="_self">Th21</a>, <a class="txt" href="jordan4.html#E10:31"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan4.html#E12:31"><i><font color="Green">E57</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E9:31_3_1_3_1_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E10:31_3_1_3_1_1_2_2_1_1_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <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="E11:31_3_1_3_1_1_2_2_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_3_1_1_2_2_1_2"><b>case </b></a><a NAME="E1:31_3_1_3_1_1_2_2_1_2"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:31_3_1_3_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, </i>;<br/><a NAME="E3:31_3_1_3_1_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan4.html#T21" target="_self">Th21</a>, <a class="ref" href="jordan4.html#T24" target="_self">Th24</a></i>;<br/><b>hence </b><a NAME="E4:31_3_1_3_1_1_2_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan4.html#E2:31"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E10:31_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E129</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 <b>by </b><i><a class="txt" href="jordan4.html#E2:31"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, , </i>;<br/><a NAME="E11:31_3_1_3_1_1_2"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3_1_3_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:31_3_1_3_1_1_2_3"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 ;<br/>

<a NAME="E2:31_3_1_3_1_1_2_3"/><b>then </b><i><font color="Green">E130</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:31_3_1_3_1_1_2_3"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan4.html#E2:31"><i><font color="Green">E33</font></i></a>, <a class="ref" href="jordan4.html#T1" target="_self">Th1</a></i>;<br/>
<i><font color="Green">E132</font></i>: <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="jordan4.html#E2:31"><i><font color="Green">E33</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> 1</span>)</span>

;<br/>
<a NAME="E5:31_3_1_3_1_1_2_3"/>
1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E6:31_3_1_3_1_1_2_3"/><b>then </b>
<font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 0 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/>
<a NAME="E7:31_3_1_3_1_1_2_3"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font>
 
<b>by </b><i><a class="ref" href="jordan4.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan4.html#T25" target="_self">Th25</a>, <a class="txt" href="jordan4.html#E10:31"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:31_3_1_3_1_1_2_3"/><b>then </b><i><font color="Green">E134</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E9:31_3_1_3_1_1_2_3"/>
<font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="jordan4.html#E11:31"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E10:31_3_1_3_1_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E11:31_3_1_3_1_1_2_3"/><b>then </b><i><font color="Green">E135</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3_1_3_1_1_2_3_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_3_1_1_2_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_3_1_3_1_1_2_3_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_3_1_1_2_3_2_1_1"><b>case </b></a><a NAME="E1:31_3_1_3_1_1_2_3_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
<b>by </b><i><a class="txt" href="jordan4.html#E11:31"><i><font color="Green">E55</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="txt" href="jordan4.html#E5:31"><i><font color="Green">E35</font></i></a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>
;<br/><a NAME="E3:31_3_1_3_1_1_2_3_2_1_1"/><b>then </b>
<font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/><b>hence </b><a NAME="E4:31_3_1_3_1_1_2_3_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan4.html#E11:31"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_3_1_1_2_3_2_1_2"><b>case </b></a><a NAME="E1:31_3_1_3_1_1_2_3_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 ;<br/><div class="add"><a NAME="E2:31_3_1_3_1_1_2_3_2_1_2"/><i><font color="Green">E136</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E11:31"><i><font color="Green">E55</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3_1_3_1_1_2_3_2_1_2_1"><b>now </b></a><div class="add"><a NAME="E1:31_3_1_3_1_1_2_3_2_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a>, <a class="ref" href="jordan4.html#T25" target="_self">Th25</a></i>;<br/><a NAME="E2:31_3_1_3_1_1_2_3_2_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E3:31_3_1_3_1_1_2_3_2_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E4:31_3_1_3_1_1_2_3_2_1_2_1"/><b>then </b>
<font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E5:31_3_1_3_1_1_2_3_2_1_2_1"/><b>then </b>
<font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan4.html#T7" target="_self">Th7</a>, <a class="txt" href="jordan4.html#E11:31"><i><font color="Green">E55</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E6:31_3_1_3_1_1_2_3_2_1_2_1"/><b>then </b><i><font color="Green">E137</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <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="31_3_1_3_1_1_2_3_2_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_3_1_1_2_3_2_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="E1:31_3_1_3_1_1_2_3_2_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 or <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 )
 </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="31_3_1_3_1_1_2_3_2_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:31_3_1_3_1_1_2_3_2_1_2_1_1_1_1"/><i><font color="Green">E138</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 ;<br/><div class="add"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="jordan4.html#E11:31"><i><font color="Green">E55</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan4.html#T7" target="_self">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan4.html#E11:31"><i><font color="Green">E55</font></i></a>, <a class="ref" href="jordan4.html#T27" target="_self">Th27</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/><a NAME="E3:31_3_1_3_1_1_2_3_2_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:31_3_1_3_1_1_2_3_2_1_2_1_1_1_1"/>
 <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="jordan4.html#T27" target="_self">Th27</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_3_1_1_2_3_2_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:31_3_1_3_1_1_2_3_2_1_2_1_1_1_2"/><i><font color="Green">E139</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 ;<br/><div class="add"><i><font color="Green">E140</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="jordan4.html#E5:31"><i><font color="Green">E35</font></i></a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="jordan4.html#E11:31"><i><font color="Green">E55</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan4.html#E11:31"><i><font color="Green">E55</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan4.html#T28" target="_self">Th28</a>, <a class="ref" href="nat_1.html#T74">NAT_1:74</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="nat_1.html#T77">NAT_1:77</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="jordan4.html#T27" target="_self">Th27</a>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/><a NAME="E5:31_3_1_3_1_1_2_3_2_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E141</font></i>: 
 <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E12:31"><i><font color="Green">E57</font></i></a>, </i>;<br/><a NAME="E6:31_3_1_3_1_1_2_3_2_1_2_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
 ;<br/><a NAME="E7:31_3_1_3_1_1_2_3_2_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E142</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E5:31"><i><font color="Green">E35</font></i></a>, , , <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E8:31_3_1_3_1_1_2_3_2_1_2_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T77">NAT_1:77</a></i>;<br/><b>then </b> <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan4.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan4.html#T30" target="_self">Th30</a>, <a class="ref" href="nat_1.html#T74">NAT_1:74</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>

;<br/><b>hence </b><a NAME="E10:31_3_1_3_1_1_2_3_2_1_2_1_1_1_2"/>
 <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="jordan4.html#T27" target="_self">Th27</a>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:31_3_1_3_1_1_2_3_2_1_2_1"/>
 <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:31_3_1_3_1_1_2_3_2_1_2"/>
 <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E13:31_3_1_3_1_1_2_3"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E2:31"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan4.html#E4:31"><i><font color="Green">E34</font></i></a>, , <a class="ref" href="jordan4.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan4.html#T25" target="_self">Th25</a>, <a class="txt" href="jordan4.html#E9:31"><i><font color="Green">E53</font></i></a>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E12:31_3_1_3_1_1_2"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font> )
 <b>by </b><i><a class="ref" href="jordan4.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan4.html#T24" target="_self">Th24</a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:31_3_1_3"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E18:31"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font> )
 ;<br/>


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