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<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L1">multLoopStr</a> , <font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L1">multLoopStr</a> ;<br/><b>let </b><font color="Maroon">F</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">lc</font> be    <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <font color="Maroon">F</font>;<br/><b>let </b><font color="Maroon">G</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">E</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K3">:]</a></span>;<br/>











<b>assume </b><a NAME="E1:65"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 ;<br/>

<b>assume </b><a NAME="E2:65"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">E</font> <a href="ideal_1.html#R1" target="_self">represents</a> <font color="Maroon">lc</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">E</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">E</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">E</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">LC</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E4:65"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">LC</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font>
 <b>and </b><br/><a NAME="E5:65"/><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="finseq_1.html#K3">len</a> <font color="Maroon">lc</font></span>)</span> holds <br/><font color="Maroon">LC</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_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:65_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">LC</font>
 ;<br/><a NAME="E2:65_1"/><i><font color="Green">E76</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">LC</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">lc</font></span>)</span>
 <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">u</font> = <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>, <font color="Maroon">v</font> = <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ;<br/><a NAME="E4:65_1"/><i><font color="Green">E77</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:65_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">lc</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">LC</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E6:65_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#D1" target="_self">Def1</a>, <a class="ref" href="ideal_1.html#D2" target="_self">Def2</a>, </i>;<br/><a NAME="E7:65_1"/><b>then </b>
<font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#D3" target="_self">Def3</a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/><b>take </b><font color="Maroon">u</font> = <font color="Maroon">u</font>;<br/><b>take </b><font color="Maroon">v</font> = <font color="Maroon">v</font>;<br/><b>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><a NAME="E9:65_1"/>
<font color="Maroon">LC</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LC</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#D2" target="_self">Def2</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E10:65_1"/>
<font color="Maroon">LC</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="ideal_1.html#D2" target="_self">Def2</a>, </i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">LC</font> = <font color="Maroon">LC</font> as    <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">LC</font>
;<br/>

<b>thus </b><a NAME="E8:65"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">LC</font>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E9:65"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">LC</font>
 ;<br/>

<a NAME="E10:65"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">LC</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">lc</font></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="E11:65"/>
<font color="Maroon">LC</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">E</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">E</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">E</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="ideal_1.html#D2" target="_self">Def2</a></i>;<br/>


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