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<a NAME="E1:108_1"/><i><font color="Green" title="E52">A2</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon">F<sub>2</sub></font>() holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">x</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:108_1_1"/><i><font color="Green" title="E53">A3</font></i>: 
<font color="Maroon" title="c1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon">F<sub>2</sub></font>()
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c2">k</font> = <font color="Maroon" title="c1">y</font> as    <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>consider </b><font color="Maroon" title="c3">x</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="E4:108_1_1"/><i><font color="Green" title="E54">A4</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">x</font>]
 <b>by </b><i><a class="txt" href="finseq_1.html#E1:108"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="finseq_1.html#E1:108_1_1"><i><font color="Green" title="E53">A3</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">x</font>
;<br/>

<b>thus </b><a NAME="E5:108_1_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">y</font>,<font color="Maroon" title="c3">x</font>] )
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:108_1_1"><i><font color="Green" title="E54">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c1">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:108_1"/><i><font color="Green" title="E53">A5</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>and </b><br/><a NAME="E4:108_1"/><i><font color="Green" title="E54">A6</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E5:108_1"/><i><font color="Green" title="E55">A7</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="finseq_1.html#E1:108_1"><i><font color="Green" title="E52">A2</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c2">f</font> = <font color="Maroon" title="c1">f</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="txt" href="finseq_1.html#E3:108_1"><i><font color="Green" title="E53">A5</font></i></a>, <a class="ref" href="finseq_1.html#D2" target="_self" title="FINSEQ_1:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">p</font> = <font color="Maroon" title="c2">f</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="finseq_1.html#E4:108_1"><i><font color="Green" title="E54">A6</font></i></a>, <a class="ref" href="finseq_1.html#D4" target="_self" title="FINSEQ_1:def.4">Def4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">p</font>
;<br/>

<b>thus </b><a NAME="E8:108_1"/>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive" title="b1">k</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="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>] ) )
 <b>by </b><i><a class="txt" href="finseq_1.html#E3:108_1"><i><font color="Green" title="E53">A5</font></i></a>, <a class="txt" href="finseq_1.html#E5:108_1"><i><font color="Green" title="E55">A7</font></i></a></i>;<br/>


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