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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g</font> &amp;  <a href="finseq_8.html#K2" title="FINSEQ_8:func.2">smid</a> <font color="Maroon" title="c3">g</font>,1,$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_8.html#K2" title="FINSEQ_8:func.2">smid</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> $1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> );<br/>
<a NAME="E1:12_1_1"/><i><font color="Green" title="E7">A1</font></i>: 
 <a href="finseq_8.html#K2" title="FINSEQ_8:func.2">smid</a> <font color="Maroon" title="c3">g</font>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_8.html#T7" target="_self" title="FINSEQ_8:th.7">Th7</a></i>;<br/>
<a NAME="E2:12_1_1"/><i><font color="Green" title="E8">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1

;<br/>
<a NAME="E4:12_1_1"/><b>then </b>
 <a href="finseq_8.html#K2" title="FINSEQ_8:func.2">smid</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="finseq_8.html#E2:12_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="finseq_8.html#T7" target="_self" title="FINSEQ_8:th.7">Th7</a></i>;<br/>
<a NAME="E5:12_1_1"/><b>then </b><i><font color="Green" title="E9">A3</font></i>: 
 ex <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>]
 
<b>by </b><i><a class="txt" href="finseq_8.html#E1:12_1_1"><i><font color="Green" title="E7">A1</font></i></a></i>;<br/>
<a NAME="E6:12_1_1"/><i><font color="Green" title="E10">A4</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g</font>
 
;<br/>
<b>consider </b><font color="Maroon" title="c4">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E8:12_1_1"/><i><font color="Green" title="E11">A5</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="finseq_8.html#E6:12_1_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="finseq_8.html#E5:12_1_1"><i><font color="Green" title="E9">A3</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c5">k</font> = <font color="Maroon" title="c4">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">b</font> =  <a href="finseq_8.html#K2" title="FINSEQ_8:func.2">smid</a> <font color="Maroon" title="c3">g</font>,1,<font color="Maroon" title="c5">k</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_1_2_1"/>
( <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2_1_1"><b>case </b></a><a NAME="E1:12_1_1_2_1_1"/><i><font color="Green" title="E12">A6</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:12_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E13">A7</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:12_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E14">A8</font></i>: 
 <a href="finseq_8.html#K2" title="FINSEQ_8:func.2">smid</a> <font color="Maroon" title="c3">g</font>,1,<font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">g</font>,1,<font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="ref" href="finseq_8.html#T4" target="_self" title="FINSEQ_8:th.4">Th4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_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="E1:12_1_1_2_1_1_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:12_1_1_2_1_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:12_1_1_2_1_1_1_1_1"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K2" title="FINSEQ_8:func.2">smid</a> <font color="Maroon" title="c3">g</font>,1,<font color="Maroon" title="c5">k</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="txt" href="finseq_8.html#E8:12_1_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="txt" href="finseq_8.html#E2:12_1_1_2_1_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="finseq_8.html#E3:12_1_1_2_1_1"><i><font color="Green" title="E14">A8</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<b>by </b><i><a class="txt" href="finseq_8.html#E2:12_1_1_2_1_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon" title="c5">k</font>

;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:12_1_1_2_1_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_1_1_2_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="finseq_8.html#E8:12_1_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="txt" href="finseq_8.html#E1:12_1_1_2_1_1"><i><font color="Green" title="E12">A6</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:12_1_1_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K2" title="FINSEQ_8:func.2">smid</a> <font color="Maroon" title="c3">g</font>,1,<font color="Maroon" title="c5">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2_1_2"><b>case </b></a><a NAME="E1:12_1_1_2_1_2"/><i><font color="Green" title="E12">A9</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:12_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E13">A10</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E3:12_1_1_2_1_2"/>
 <a href="finseq_8.html#K2" title="FINSEQ_8:func.2">smid</a> <font color="Maroon" title="c3">g</font>,1,<font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="finseq_8.html#E1:12_1_1_2_1_2"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="finseq_8.html#T7" target="_self" title="FINSEQ_8:th.7">Th7</a></i>;<br/><b>hence </b><a NAME="E4:12_1_1_2_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K2" title="FINSEQ_8:func.2">smid</a> <font color="Maroon" title="c3">g</font>,1,<font color="Maroon" title="c5">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="finseq_8.html#E2:12_1_1_2_1_2"><i><font color="Green" title="E13">A10</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:12_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_8.html#K2" title="FINSEQ_8:func.2">smid</a> <font color="Maroon" title="c3">g</font>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_8.html#K2" title="FINSEQ_8:func.2">smid</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; ( for <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g</font> &amp;  <a href="finseq_8.html#K2" title="FINSEQ_8:func.2">smid</a> <font color="Maroon" title="c3">g</font>,1,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_8.html#K2" title="FINSEQ_8:func.2">smid</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> holds <br/><font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="finseq_8.html#E8:12_1_1"><i><font color="Green" title="E11">A5</font></i></a></i>;<br/>


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