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<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> ;<br/><b>let </b><font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><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">g1</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">g2</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>assume </b><a NAME="E1:59"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">g1</font> <a href="jordan4.html#R1" target="_self">is_a_part&gt;_of</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">g2</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> )
 ;<br/>

<a NAME="E2:59"/><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">g1</font> <a href="jordan4.html#R1" target="_self">is_a_part&gt;_of</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">g2</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> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:59"/><i><font color="Green">E34</font></i>: 
1 <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="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:59"/><b>then </b><i><font color="Green">E35</font></i>: 
1 <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="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:59"/><b>then </b><i><font color="Green">E43</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="binarith.html#K3">-'</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="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E6:59"/><i><font color="Green">E44</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="binarith.html#K3">-'</a> 1 <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="binarith.html#K3">-'</a> 1</span>)</span> <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="E7:59"/><b>then </b><i><font color="Green">E53</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="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#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E8:59"/><i><font color="Green">E54</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="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E9:59"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">i1</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="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_1"/>
( <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> or <font color="Maroon">i1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1"><b>case </b></a><a NAME="E1:59_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_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="E1:59_1_1_1_1_1"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> or <font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</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="59_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:59_1_1_1_1_1_1"/>
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:59_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:59_1_1_1_1_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">g2</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>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<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>,<font color="Maroon">i2</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59_1_1_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan4.html#E2:59"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E3:59_1_1_1_1_1_2"/>
<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> 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#E4:59_1_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E4:59_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E59</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#E4:59_1_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="jordan4.html#T29" target="_self">Th29</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:59_1_1_1_1_1_2"/><i><font color="Green">E68</font></i>: 
 <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>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <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#E4:59_1_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="jordan4.html#T31" target="_self">Th31</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_1_1_1_1_2_1_1"/>
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> or 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> )
 </span><b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2"><i><font color="Green">E59</font></i></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="59_1_1_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:59_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E78</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font>
 ;<br/><div class="add"><a NAME="E2:59_1_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:59_1_1_1_1_1_2_1_1_1"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="jordan4.html#T34" target="_self">Th34</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:59_1_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a>  <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>,1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="jordan4.html#T33" target="_self">Th33</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><i><font color="Green">E81</font></i>: <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<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>,<font color="Maroon">i2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 2
<b>by </b><i><a class="txt" href="jordan4.html#E2:59_1_1_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan4.html#E5:59_1_1_1_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan4.html#E2:59"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,1</span>)</span> <a href="funct_1.html#K1">.</a> 2
<b>by </b><i><a class="ref" href="jordan4.html#D2" target="_self">Def2</a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan4.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan4.html#T31" target="_self">Th31</a>, <a class="ref" href="jordan4.html#T34" target="_self">Th34</a>, <a class="ref" href="jordan4.html#D2" target="_self">Def2</a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</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> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan4.html#T35" target="_self">Th35</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1</span>)</span>

<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/><a NAME="E6:59_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E5:59_1_1_1_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E7:59_1_1_1_1_1_2_1_1_1"/><b>then </b>
<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> <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:59_1_1_1_1_1_2_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E9:59_1_1_1_1_1_2_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i1</font></span>)</span> <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="E10:59_1_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a>  <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>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="jordan4.html#T31" target="_self">Th31</a>, <a class="txt" href="jordan4.html#E5:59_1_1_1_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><i><font color="Green">E83</font></i>: <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 2 = 
<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> 2
<b>by </b><i><a class="txt" href="jordan4.html#E2:59_1_1_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan4.html#E5:59_1_1_1_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="jordan4.html#T3" target="_self">Th3</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="jordan4.html#T31" target="_self">Th31</a>, <a class="txt" href="jordan4.html#E5:59_1_1_1_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="jordan4.html#D4" target="_self">Def4</a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <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> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/><a NAME="E12:59_1_1_1_1_1_2_1_1_1"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="jordan4.html#T34" target="_self">Th34</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E13:59_1_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default">1 <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> <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E14:59_1_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><div><i><font color="Green">E86</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1_1_1_2_1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_1_1_1_1_2_1_1_1_3"/><i><font color="Green">E87</font></i>: 
( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 &amp; <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 ;<br/><a NAME="E2:59_1_1_1_1_1_2_1_1_1_3"/><b>then </b>
1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><b>hence </b><a NAME="E3:59_1_1_1_1_1_2_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="goboard7.html#T36">GOBOARD7:36</a>, <a class="txt" href="jordan4.html#E5:59_1_1_1_1_1_2_1_1_1"><i><font color="Green">E81</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1_1_1_2_1_1_1_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1_1_2_1_1_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_1_1_1_1_2_1_1_1_4_1"/>
( 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 or <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 </span><b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2_1_1_1"><i><font color="Green">E80</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1_1_2_1_1_1_4_1_1"><b>case </b></a><a NAME="E1:59_1_1_1_1_1_2_1_1_1_4_1_1"/><i><font color="Green">E88</font></i>: 
1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1
 ;<br/><div class="add"><a NAME="E2:59_1_1_1_1_1_2_1_1_1_4_1_1"/>
<font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <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="E3:59_1_1_1_1_1_2_1_1_1_4_1_1"/><b>then </b>
<font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E4:59_1_1_1_1_1_2_1_1_1_4_1_1"/><b>then </b><i><font color="Green">E89</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 &amp; <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 &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> )
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59_1_1_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan4.html#E2:59_1_1_1_1_1_2_1_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan4.html#E5:59_1_1_1_1_1_2_1_1_1"><i><font color="Green">E81</font></i></a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:59_1_1_1_1_1_2_1_1_1_4_1_1"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <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"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="goboard7.html#T39">GOBOARD7:39</a></i>;<br/><a NAME="E6:59_1_1_1_1_1_2_1_1_1_4_1_1"/>
<font color="Maroon">i1</font> <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="txt" href="jordan4.html#E10:59_1_1_1_1_1_2_1_1_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:59_1_1_1_1_1_2_1_1_1_4_1_1"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59_1_1_1_1_1_2_1_1_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E8:59_1_1_1_1_1_2_1_1_1_4_1_1"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E9:59_1_1_1_1_1_2_1_1_1_4_1_1"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 2
 <b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="jordan4.html#D3" target="_self">Def3</a>, <a class="ref" href="jordan4.html#T37" target="_self">Th37</a>, <a class="txt" href="jordan4.html#E11:59_1_1_1_1_1_2_1_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan4.html#E13:59_1_1_1_1_1_2_1_1_1"><i><font color="Green">E84</font></i></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="59_1_1_1_1_1_2_1_1_1_4_1_2"><b>case </b></a><a NAME="E1:59_1_1_1_1_1_2_1_1_1_4_1_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:59_1_1_1_1_1_2_1_1_1_4_1_2"/>
<font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <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="E3:59_1_1_1_1_1_2_1_1_1_4_1_2"/><b>then </b>
<font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E4:59_1_1_1_1_1_2_1_1_1_4_1_2"/><b>then </b><i><font color="Green">E93</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 &amp; <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</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="txt" href="jordan4.html#E4:59_1_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_1_1_1_2_1_1_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jordan4.html#E5:59_1_1_1_1_1_2_1_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:59_1_1_1_1_1_2_1_1_1_4_1_2"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <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"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="goboard7.html#T38">GOBOARD7:38</a></i>;<br/><a NAME="E6:59_1_1_1_1_1_2_1_1_1_4_1_2"/>
<font color="Maroon">i1</font> <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="txt" href="jordan4.html#E10:59_1_1_1_1_1_2_1_1_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:59_1_1_1_1_1_2_1_1_1_4_1_2"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E10:59_1_1_1_1_1_2_1_1_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E8:59_1_1_1_1_1_2_1_1_1_4_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E9:59_1_1_1_1_1_2_1_1_1_4_1_2"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 2
 <b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="jordan4.html#D3" target="_self">Def3</a>, <a class="ref" href="jordan4.html#T37" target="_self">Th37</a>, <a class="txt" href="jordan4.html#E11:59_1_1_1_1_1_2_1_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan4.html#E13:59_1_1_1_1_1_2_1_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E17:59_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 2
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:59_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E96</font></i>: 
1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font>
 ;<br/><div class="add"><i><font color="Green">E97</font></i>:  <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>,1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <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#E4:59_1_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="jordan4.html#T31" target="_self">Th31</a>, </i>
<br/>.= 
0 <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="jordan4.html#T34" target="_self">Th34</a>, <a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
1

;<br/><i><font color="Green">E98</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</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="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i2</font>
<b>by </b><i><a class="txt" href="jordan4.html#E2:59_1_1_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan4.html#E5:59_1_1_1_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan4.html#E2:59"><i><font color="Green">E33</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="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="jordan4.html#T34" target="_self">Th34</a>, </i>
;<br/><a NAME="E4:59_1_1_1_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">i2</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> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:59_1_1_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">i2</font></span>)</span> <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"><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> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E6:59_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E99</font></i>: 
1 <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">g2</font>
 <b>by </b><i><a class="ref" href="jordan4.html#D2" target="_self">Def2</a></i>;<br/><i><font color="Green">E100</font></i>:  <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>,1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <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#E4:59_1_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="jordan4.html#T31" target="_self">Th31</a>, </i>
<br/>.= 
0 <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="jordan4.html#T34" target="_self">Th34</a>, <a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
1

;<br/><i><font color="Green">E101</font></i>: <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 2 = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<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>,<font color="Maroon">i2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan4.html#T32" target="_self">Th32</a>, <a class="ref" href="jordan4.html#T35" target="_self">Th35</a>, <a class="ref" href="jordan4.html#D3" target="_self">Def3</a>, <a class="ref" href="jordan3.html#T15">JORDAN3:15</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_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#E4:59_1_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, <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="jordan4.html#D1" target="_self">Def1</a>, <a class="ref" href="jordan4.html#D4" target="_self">Def4</a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
;<br/><a NAME="E9:59_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E5:59_1_1_1_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:59_1_1_1_1_1_2_1_1_2"/><b>then </b>
<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> <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E11:59_1_1_1_1_1_2_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E12:59_1_1_1_1_1_2_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i1</font></span>)</span> <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="E13:59_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E102</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a>  <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>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="jordan4.html#T31" target="_self">Th31</a>, <a class="txt" href="jordan4.html#E5:59_1_1_1_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><i><font color="Green">E103</font></i>: <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 2 = 
<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> 2
<b>by </b><i><a class="txt" href="jordan4.html#E2:59_1_1_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan4.html#E5:59_1_1_1_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="jordan4.html#T3" target="_self">Th3</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="jordan4.html#T31" target="_self">Th31</a>, <a class="txt" href="jordan4.html#E5:59_1_1_1_1_1_2"><i><font color="Green">E68</font></i></a>, , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <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> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/><a NAME="E15:59_1_1_1_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 4
 <b>by </b><i><a class="ref" href="goboard7.html#T36">GOBOARD7:36</a></i>;<br/><a NAME="E16:59_1_1_1_1_1_2_1_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="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E17:59_1_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">i1</font> <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
 <b>by </b><i><a class="ref" href="jordan4.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan4.html#T34" target="_self">Th34</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E18:59_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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"><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="jordan4.html#T34" target="_self">Th34</a>, <a class="ref" href="goboard7.html#T39">GOBOARD7:39</a></i>;<br/><a NAME="E19:59_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">f</font> <a href="funct_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> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</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="jordan4.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E20:59_1_1_1_1_1_2_1_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E21:59_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 2
 <b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="jordan4.html#T37" target="_self">Th37</a>, , , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:59_1_1_1_1_1_2"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 2
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1_1_1"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 2
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_2"><b>case </b></a><a NAME="E1:59_1_1_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_1_2"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">g1</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>,<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="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,1,<font color="Maroon">i2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan4.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E3:59_1_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:59_1_1_2"/><b>then </b><i><font color="Green">E108</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:59_1_1_2"/><i><font color="Green">E109</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> 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="E6:59_1_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="ref" href="jordan4.html#T29" target="_self">Th29</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:59_1_1_2"/><b>then </b><i><font color="Green">E110</font></i>: 
 <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>,<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="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="binarith.html#K3">-'</a> <font color="Maroon">i1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <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="jordan4.html#T31" target="_self">Th31</a>, <a class="ref" href="jordan4.html#D1" target="_self">Def1</a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E8:59_1_1_2"/><i><font color="Green">E111</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> 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/><div><i><font color="Green">E112</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_1_2_1_1"/>
( <font color="Maroon">i1</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 or <font color="Maroon">i1</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="xcmplx_0.html#K6">-</a> 1 )
 </span><b>by </b><i><a class="ref" href="jordan4.html#D1" target="_self">Def1</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="59_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:59_1_1_2_1_1_1"/>
<font color="Maroon">i1</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/><div class="add"><a NAME="E2:59_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">i1</font> <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="jordan4.html#T29" target="_self">Th29</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:59_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E113</font></i>: 
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 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>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:59_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E114</font></i>: 
<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="real_1.html#K5">-</a> <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</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#K5">-</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E5:59_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <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">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</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#K5">-</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="jordan4.html#D2" target="_self">Def2</a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E6:59_1_1_2_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K3">+</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="real_1.html#K5">-</a> <font color="Maroon">i1</font></span>)</span>
 ;<br/><a NAME="E7:59_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E115</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a>  <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>,<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="ref" href="jordan4.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan4.html#T33" target="_self">Th33</a>, <a class="ref" href="jordan4.html#T34" target="_self">Th34</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><b>thus </b><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<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="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,1,<font color="Maroon">i2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 2
<b>by </b><i>, , <a class="ref" href="jordan4.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<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="funct_1.html#K1">.</a> 2
<b>by </b><i><a class="ref" href="jordan4.html#D4" target="_self">Def4</a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <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> 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="jordan4.html#T34" target="_self">Th34</a>, <a class="ref" href="jordan4.html#D3" target="_self">Def3</a>, <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:59_1_1_2_1_1_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">i1</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="xcmplx_0.html#K6">-</a> 1
 ;<br/><div class="add"><b>then </b><i><font color="Green">E117</font></i>:  <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>,<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> = 
<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="binarith.html#K3">-'</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="nat_1.html#K1">+</a> 1
<b>by </b><i>, <a class="ref" href="jordan4.html#T29" target="_self">Th29</a>, , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
<br/>.= 
0 <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
1

;<br/><i><font color="Green">E118</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</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">i2</font></span>)</span> <a href="binarith.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="ref" href="jordan4.html#T29" target="_self">Th29</a>, , <a class="ref" href="jordan4.html#D2" target="_self">Def2</a>, <a class="ref" href="jordan4.html#T6" target="_self">Th6</a></i>
<br/>.= 
<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="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="binarith.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>

<br/>.= 
1 <a href="nat_1.html#K1">+</a> <font color="Maroon">i2</font>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/><a NAME="E4:59_1_1_2_1_1_2"/><i><font color="Green">E119</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <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/><i><font color="Green">E120</font></i>:  <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>,<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> = 
<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="binarith.html#K3">-'</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="nat_1.html#K1">+</a> 1
<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="jordan4.html#D2" target="_self">Def2</a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
<br/>.= 
0 <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
1

;<br/><b>thus </b><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 2 = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,1,<font color="Maroon">i2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<span class="p4">(<span class="default"><span class="p5">(<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></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="jordan4.html#D3" target="_self">Def3</a>, <a class="ref" href="jordan4.html#D4" target="_self">Def4</a>, <a class="ref" href="jordan4.html#T37" target="_self">Th37</a>, <a class="ref" href="jordan3.html#T15">JORDAN3:15</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i>, <a class="ref" href="jordan4.html#T28" target="_self">Th28</a>, , , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan4.html#T27" target="_self">Th27</a>, <a class="ref" href="jordan4.html#D2" target="_self">Def2</a>, </i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E10:59_1_1_2"/>
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E11:59_1_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">i2</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E12:59_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E13:59_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <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="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E14:59_1_1_2"/><b>then </b><i><font color="Green">E121</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a>  <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>
 <b>by </b><i>, , <a class="ref" href="jordan4.html#T31" target="_self">Th31</a>, , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><i><font color="Green">E122</font></i>: <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 2 = 
<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> 2
<b>by </b><i>, , <a class="txt" href="jordan4.html#E1:59"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="jordan4.html#T31" target="_self">Th31</a>, , <a class="ref" href="jordan4.html#D2" target="_self">Def2</a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</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> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan4.html#T32" target="_self">Th32</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1</span>)</span>

<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/><a NAME="E16:59_1_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E17:59_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E18:59_1_1_2"/><b>then </b><i><font color="Green">E123</font></i>: 
<span class="p1">(<span class="default">1 <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> <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E19:59_1_1_2"/><b>then </b><i><font color="Green">E124</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><div><i><font color="Green">E126</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_1_2_3"/><i><font color="Green">E127</font></i>: 
( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 &amp; <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 ;<br/><a NAME="E2:59_1_1_2_3"/><b>then </b>
1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><b>hence </b><a NAME="E3:59_1_1_2_3"/>
contradiction
 <b>by </b><i><a class="ref" href="goboard7.html#T36">GOBOARD7:36</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_2_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_1_2_4_1"/>
( 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 or <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_2_4_1_1"><b>case </b></a><a NAME="E1:59_1_1_2_4_1_1"/><i><font color="Green">E128</font></i>: 
1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1
 ;<br/><div class="add"><a NAME="E2:59_1_1_2_4_1_1"/>
<font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <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="E3:59_1_1_2_4_1_1"/><b>then </b>
<font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E4:59_1_1_2_4_1_1"/><b>then </b><i><font color="Green">E129</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 &amp; <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 &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> )
 <b>by </b><i>, <a class="ref" href="jordan4.html#T37" target="_self">Th37</a>, , , <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:59_1_1_2_4_1_1"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <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"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="goboard7.html#T39">GOBOARD7:39</a></i>;<br/><a NAME="E6:59_1_1_2_4_1_1"/>
<font color="Maroon">i1</font> <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="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:59_1_1_2_4_1_1"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan4.html#T37" target="_self">Th37</a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E8:59_1_1_2_4_1_1"/>
1 <a href="xxreal_0.html#NR3" target="_self">&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#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E9:59_1_1_2_4_1_1"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 2
 <b>by </b><i>, <a class="ref" href="jordan4.html#T35" target="_self">Th35</a>, <a class="ref" href="jordan4.html#D3" target="_self">Def3</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="59_1_1_2_4_1_2"><b>case </b></a><a NAME="E1:59_1_1_2_4_1_2"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:59_1_1_2_4_1_2"/>
<font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <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="E3:59_1_1_2_4_1_2"/><b>then </b>
<font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E4:59_1_1_2_4_1_2"/><b>then </b><i><font color="Green">E133</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 &amp; <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</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#D4" target="_self">Def4</a>, , <a class="ref" href="jordan3.html#T1">JORDAN3:1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:59_1_1_2_4_1_2"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <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"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="goboard7.html#T38">GOBOARD7:38</a></i>;<br/><a NAME="E6:59_1_1_2_4_1_2"/>
<font color="Maroon">i1</font> <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="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:59_1_1_2_4_1_2"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E8:59_1_1_2_4_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&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#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E9:59_1_1_2_4_1_2"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 2
 <b>by </b><i>, <a class="ref" href="jordan4.html#T35" target="_self">Th35</a>, <a class="ref" href="jordan4.html#D3" target="_self">Def3</a>, , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E22:59_1_1_2"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 2
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:59"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 2
 ;<br/>


</div>
