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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon">F<sub>1</sub></font>()  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>];<br/>
<a NAME="E1:26_1"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="finseq_2.html#E1:26"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></i>;<br/>
<a NAME="E2:26_1"/><i><font color="Green" title="E23">A4</font></i>: 
for <font color="Olive" title="b1">i</font> being   <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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">i</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:26_1_1"/><i><font color="Green" title="E24">A5</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon">F<sub>1</sub></font>()  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c2">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E2:26_1_1"/><i><font color="Green" title="E25">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">q</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon" title="c4">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E4:26_1_1"/><i><font color="Green" title="E26">A7</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T22" target="_self" title="FINSEQ_2:th.22">Th22</a></i>;<br/>
<a NAME="E5:26_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="finseq_2.html#E4:26_1_1"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="finseq_2.html#T19" target="_self" title="FINSEQ_2:th.19">Th19</a></i>;<br/>
<a NAME="E6:26_1_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">q</font>]
 
<b>by </b><i><a class="txt" href="finseq_2.html#E1:26_1_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="finseq_2.html#E2:26_1_1"><i><font color="Green" title="E25">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:26_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>]
 <b>by </b><i><a class="txt" href="finseq_2.html#E2:26"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="finseq_2.html#E4:26_1_1"><i><font color="Green" title="E26">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:26_1"/><i><font color="Green" title="E24">A8</font></i>: 
for <font color="Olive" title="b1">i</font> being   <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="finseq_2.html#E1:26_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="finseq_2.html#E2:26_1"><i><font color="Green" title="E23">A4</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c1">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<a NAME="E4:26_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font>
 
;<br/>
<b>hence </b><a NAME="E5:26_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>]
 <b>by </b><i><a class="txt" href="finseq_2.html#E3:26_1"><i><font color="Green" title="E24">A8</font></i></a></i>;<br/>


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