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<b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">F<sub>1</sub></font>() as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>] &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() );<br/>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1"><b>now </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="E1:4_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">F<sub>2</sub></font>()
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>consider </b><font color="Maroon">d</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="E4:4_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x'</font>,<font color="Maroon">d</font>]
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">d</font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">y</font>;<br/><b>thus </b><a NAME="E6:4_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>] )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:4_1"/><i><font color="Green">E43</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">F<sub>2</sub></font>() &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; ( 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>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i>, <a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E7:4_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E8:4_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">F<sub>2</sub></font>()
 ;<br/>

<a NAME="E9:4_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:4_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>]
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


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