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<a NAME="E1:84_1_1"/><i><font color="Green">E42</font></i>: 
(  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a>, <a class="ref" href="polynom1.html#T41">POLYNOM1:41</a></i>;<br/>
<a NAME="E2:84_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">s</font> =  <a href="uproots.html#K1" target="_self">canFS</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span> as   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>  <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">L</font></span>)</span> =  <a href="uproots.html#K9" target="_self">fpoly_mult_root</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E5:84_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E6:84_1_1"/><i><font color="Green">E44</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="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E7:84_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">L</font></span>)</span> <a href="finseq_2.html#K3">*</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:84_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:84_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:84_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E5:84_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:84_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">i</font>)
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:84_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">L</font></span>)</span> <a href="finseq_2.html#K3">*</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">L</font></span>)</span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">it1</font> =  <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">f</font></span>)</span> 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">it1</font>
;<br/>

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

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

<b>thus </b><a NAME="E10:84_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E11:84_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K1" target="_self">canFS</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:84_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:84_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E3:84_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K9" target="_self">fpoly_mult_root</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E13:84_1_1"/>
<font color="Maroon">it1</font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>


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