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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#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="vectsp_1.html#V7">distributive</a> <a href="algstr_1.html#V1">left_zeroed</a> <a href="binom.html#V1">add-cancelable</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <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="E1:29"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  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>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
 ;<br/><a NAME="E2:29_1"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</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="E3:29_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:29_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b><span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="groeb_2.html#E1:29_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
;<br/><b>hence </b><a NAME="E5:29_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>4</sub></font> st <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>6</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#K1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:29"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> as    <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="ideal_1.html#D9">IDEAL_1:def 9</a></i>;<br/>
<a NAME="E5:29"/>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>;<br/>
<b>hence </b><a NAME="E6:29"/>
<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">-Ideal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T60">IDEAL_1:60</a></i>;<br/>


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