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<b>let </b><font color="Maroon">N</font> be  <a href="setfam_1.html#V1">with_non-empty_elements</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="ami_1.html#V2">void</a> <a href="ami_1.html#V5">IC-Ins-separated</a> <a href="ami_1.html#V8">definite</a>  <a href="ami_1.html#L1">AMI-Struct</a> of <font color="Maroon">N</font>;<br/><b>let </b><font color="Maroon">l1</font> be   <a href="ami_1.html#NM2">Instruction-Location</a> of <font color="Maroon">S</font>, <font color="Maroon">l2</font> be   <a href="ami_1.html#NM2">Instruction-Location</a> of <font color="Maroon">S</font>;<br/>





<b>assume </b><a NAME="E1:57"/><i><font color="Green">E30</font></i>: 
 <a href="amistd_1.html#K3">SUCC</a> <font color="Maroon">l1</font> <a href="hidden.html#R1">=</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">S</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">l2</font> ) );<br/>
<a NAME="E2:57"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2 holds <br/> ex <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">S</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">d</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:57_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_1"><b>suppose </b></a><a NAME="E1:57_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">l1</font>
;<br/><b>thus </b><a NAME="E2:57_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">l1</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_2"><b>suppose </b></a><a NAME="E1:57_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">l2</font>
;<br/><b>thus </b><a NAME="E2:57_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">l2</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E4:57"/><i><font color="Green">E78</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 2
 <b>and </b><br/><a NAME="E5:57"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="goboard1.html#S1">GOBOARD1:sch 1</a>(<a class="ref" href="scmpds_9.html#T6" target="_self">Th6</a>);<br/></i>
<a NAME="E6:57"/><i><font color="Green">E80</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2 &amp; 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E7:57"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 2
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:57"/><b>then </b>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">S</font> ;<br/>
<b>take </b>
<font color="Maroon">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="amistd_1.html#D8">AMISTD_1:def 8</a><br/>

<b>thus </b><a NAME="E10:57"/>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">l2</font> )
 <b>by </b><i>, , </i>;<br/>

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

<b>assume </b><a NAME="E11:57"/><i><font color="Green">E81</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<b>assume </b><a NAME="E12:57"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E13:57"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E14:57"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E15:57"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/>
<a NAME="E16:57"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E17:57"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="amistd_1.html#K3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/>


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