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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L3">doubleLoopStr</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">S</font>;<br/><b>let </b><font color="Maroon">lc</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font>;<br/>





<a NAME="E1:61"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</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>, <font color="Maroon">e</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:61"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">lc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T22">FINSEQ_2:22</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_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="E1:61_1"/><i><font color="Green">E42</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">p</font>
 ;<br/><a NAME="E2:61_1"/><i><font color="Green">E43</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">lc</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T39">FINSEQ_1:39</a></i>;<br/><b>then consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">a</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E4:61_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">lc</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">u</font> = <font color="Maroon">u</font>;<br/><b>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><b>thus </b><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">lc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
<b>by </b><i>, , <a class="ref" href="ideal_1.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as    <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

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

<b>thus </b><a NAME="E6:61"/>
<font color="Maroon">lc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i/>;<br/>

<a NAME="E7:61"/><i><font color="Green">E72</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/>
<a NAME="E8:61"/><i><font color="Green">E76</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">lc</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<b>then consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">a</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E10:61"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">lc</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>
<a NAME="E11:61"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">lc</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">lc</font> <a href="funct_1.html#K1">.</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_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D10" target="_self">IDEAL_1:def 10</a><br/>

<b>assume </b><a NAME="E12:61"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/>

<a NAME="E13:61"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/>
<a NAME="E14:61"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">u</font>
;<br/>

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

<b>thus </b><span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="ideal_1.html#D2" target="_self">Def2</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
<b>by </b><i>, , <a class="ref" href="ideal_1.html#D1" target="_self">Def1</a>, , <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
;<br/>


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