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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">a</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">a</font>] &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> );<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:7_1"/><i><font color="Green">E36</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">X</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &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>
<a NAME="E3:7_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">x</font> is   <a href="finseq_1.html#NM1">FinSequence</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_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>assume </b><a NAME="E1:7_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:7_1_1"/><b>then </b>
 ex <font color="Olive">a</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">a</font>] &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:7_1_1"/>
<font color="Maroon">x</font> is   <a href="finseq_1.html#NM1">FinSequence</a>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:7_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  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="7_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:7_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E2:7_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E3:7_1_2"/><i><font color="Green">E40</font></i>: 
 ex <font color="Olive">c</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">c</font>] &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:7_1_2"/>
 ex <font color="Olive">d</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">d</font>] &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">d</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:7_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">X</font> as   <a href="margrel1.html#NM1">relation</a> <b>by </b><i>, </i>;<br/>
<a NAME="E6:7_1"/>
for <font color="Olive">a</font> being  <a href="finseq_1.html#NM1">FinSequence</a> holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> iff ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">a</font>] ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_1_3_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E2:7_1_3_1"/><b>then </b>
 ex <font color="Olive">c</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">c</font>] &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:7_1_3_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a</font>] )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:7_1_3"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> iff ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a</font>] ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:7_1"/>
 ex <font color="Olive">r</font> being  <a href="margrel1.html#NM1">relation</a> st <br/>for <font color="Olive">a</font> being  <a href="finseq_1.html#NM1">FinSequence</a> holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">r</font> iff ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">a</font>] ) )
 ;<br/>


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