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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">F</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon">F<sub>1</sub></font>()  st  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b1">F</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">F</font>];<br/>
<a NAME="E1:58_1"/><i><font color="Green" title="E43">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:58"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="stirl2_1.html#E35"><i><font color="Green" title="E24">Lm1</font></i></a></i>;<br/>
<a NAME="E2:58_1"/><i><font color="Green" title="E44">A4</font></i>: 
for <font color="Olive" title="b1">n</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">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">n</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="E1:58_1_1"/><i><font color="Green" title="E45">A5</font></i>: 
for <font color="Olive" title="b1">F</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon">F<sub>1</sub></font>()  st  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">F</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c2">F</font> be   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E2:58_1_1"/><i><font color="Green" title="E46">A6</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 ;<br/>

<a NAME="E3:58_1_1"/><b>then </b>
<font color="Maroon" title="c2">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E35"><i><font color="Green" title="E24">Lm1</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">G</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon" title="c4">d</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="E5:58_1_1"/><i><font color="Green" title="E47">A7</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
 <b>by </b><i><a class="ref" href="stirl2_1.html#T46" target="_self" title="STIRL2_1:th.46">Th46</a></i>;<br/>
<a NAME="E6:58_1_1"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></i>;<br/>
<a NAME="E7:58_1_1"/><b>then </b>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E5:58_1_1"><i><font color="Green" title="E47">A7</font></i></a>, <a class="ref" href="afinsq_1.html#T20" title="AFINSQ_1:th.20">AFINSQ_1:20</a></i>;<br/>
<a NAME="E8:58_1_1"/><b>then </b>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E2:58_1_1"><i><font color="Green" title="E46">A6</font></i></a></i>;<br/>
<a NAME="E9:58_1_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">G</font>]
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:58_1_1"><i><font color="Green" title="E45">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:58_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">F</font>]
 <b>by </b><i><a class="txt" href="stirl2_1.html#E2:58"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="stirl2_1.html#E5:58_1_1"><i><font color="Green" title="E47">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:58_1"/><i><font color="Green" title="E45">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="stirl2_1.html#E1:58_1"><i><font color="Green" title="E43">A3</font></i></a>, <a class="txt" href="stirl2_1.html#E2:58_1"><i><font color="Green" title="E44">A4</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c1">F</font> be   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<a NAME="E4:58_1"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">F</font>
 
;<br/>
<b>hence </b><a NAME="E5:58_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">F</font>]
 <b>by </b><i><a class="txt" href="stirl2_1.html#E3:58_1"><i><font color="Green" title="E45">A8</font></i></a></i>;<br/>


</div>
