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<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> ex <font color="Olive">p</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> );<br/>
<a NAME="E1:17_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> holds <br/> ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</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">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:17_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<a NAME="E2:17_1_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> as   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<b>take </b>
<font color="Maroon">t</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font>
;<br/>

<b>take </b>
<font color="Maroon">t</font>
;<br/>

<b>thus </b><a NAME="E4:17_1_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>

<b>thus </b><a NAME="E5:17_1_1_1"/>
<font color="Maroon">t</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:17_1_1"/><i><font color="Green">E31</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>();<br/></i>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<b>thus </b><a NAME="E4:17_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<b>thus </b><a NAME="E5:17_1_1"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> holds <br/> ex <font color="Olive">p</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> )
 <b>by </b><i/>;<br/>


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