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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> <a href="finseq_1.html#K13">*</a>  &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/> <a href="finseq_1.html#K3">len</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b</font> ) );<br/>
<b>consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:17_1_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="finseq_1.html#K13">*</a> </span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">A</font>
;<br/>

<a NAME="E3:17_1_1"/>
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> iff ( <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> <a href="finseq_1.html#K13">*</a>  &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> holds <br/> <a href="finseq_1.html#K3">len</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:17_1_1_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> implies ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> <a href="finseq_1.html#K13">*</a>  &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> <a href="finseq_1.html#K3">len</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:17_1_1_1_1"/><b>then </b>
 ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> <a href="finseq_1.html#K13">*</a>  &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/> <a href="finseq_1.html#K3">len</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b</font> ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:17_1_1_1_1"/>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> <a href="finseq_1.html#K13">*</a>  &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> <a href="finseq_1.html#K3">len</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b</font> ) )
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:17_1_1_1"/>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> <a href="finseq_1.html#K13">*</a>  &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> <a href="finseq_1.html#K3">len</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b</font> ) implies <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:17_1_1"/>
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> iff ( <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> <a href="finseq_1.html#K13">*</a>  &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> holds <br/> <a href="finseq_1.html#K3">len</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b</font> ) ) )
 ;<br/>


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