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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means for <font color="Olive">b<sub>1</sub></font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of <font color="Maroon">F<sub>1</sub></font>() st  <a href="afinsq_1.html#K1">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>];<br/>
<a NAME="E1:58_1"/><i><font color="Green">E43</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">E41</font></i></a>, <a class="txt" href="stirl2_1.html#E35"><i><font color="Green">Lemma24</font></i></a></i>;<br/>
<a NAME="E2:58_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</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">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:58_1_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of <font color="Maroon">F<sub>1</sub></font>() st  <a href="afinsq_1.html#K1">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="afinsq_1.html#NM2">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">E46</font></i>: 
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1
 ;<br/>

<a NAME="E3:58_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="afinsq_1.html#K4">&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">Lemma24</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">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">E47</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="afinsq_1.html#K5">^</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="afinsq_1.html#K9">%&gt;</a></span>
 <b>by </b><i><a class="ref" href="stirl2_1.html#T46" target="_self">Th46</a></i>;<br/>
<a NAME="E6:58_1_1"/>
 <a href="afinsq_1.html#K1">len</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T38">AFINSQ_1:38</a></i>;<br/>
<a NAME="E7:58_1_1"/><b>then </b>
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E5:58_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="afinsq_1.html#T20">AFINSQ_1:20</a></i>;<br/>
<a NAME="E8:58_1_1"/><b>then </b>
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E2:58_1_1"><i><font color="Green">E46</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">c<sub>3</sub></font>]
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:58_1_1"><i><font color="Green">E45</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">c<sub>2</sub></font>]
 <b>by </b><i><a class="txt" href="stirl2_1.html#E2:58"><i><font color="Green">E42</font></i></a>, <a class="txt" href="stirl2_1.html#E5:58_1_1"><i><font color="Green">E47</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:58_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="stirl2_1.html#E1:58_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="stirl2_1.html#E2:58_1"><i><font color="Green">E44</font></i></a>);<br/></i>
<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

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


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