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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="algstr_1.html#V1">left_zeroed</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:96"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:96"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T60" target="_self">Th60</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:96_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 ;<br/><a NAME="E2:96_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T64" target="_self">Th64</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:96_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K7">*</a> <font color="Maroon">c<sub>6</sub></font>;<br/><a NAME="E5:96_1"/><i><font color="Green">E52</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K7">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D3">POLYNOM1:def 3</a></i>;<br/><a NAME="E6:96_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K7">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>4</sub></font> being   <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K7">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:96_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K7">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:96_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E5:96_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="ideal_1.html#D9" target="_self">Def9</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K7">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="ideal_1.html#E5:96_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="ideal_1.html#E1:96_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="ideal_1.html#E3:96_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="polynom1.html#D3">POLYNOM1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E5:96_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being   <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K7">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>


</div><b>end;</b></div><a NAME="E7:96_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K7">*</a> <font color="Maroon">c<sub>6</sub></font> is    <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="ideal_1.html#D9" target="_self">Def9</a></i>;<br/><a NAME="E8:96_1"/>
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K7">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:96"><i><font color="Green">E50</font></i></a>, <a class="txt" href="ideal_1.html#E4:96_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="binom.html#T5">BINOM:5</a></i>;<br/><b>hence </b><a NAME="E9:96_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E7:96_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="ideal_1.html#T60" target="_self">Th60</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:96"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div>
