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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="vectsp_2.html#NM6">LeftMod</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#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:14"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="realset2.html#V3">trivial</a>
 ;<br/>

<b>thus </b><a NAME="E2:14"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font> holds  <a href="vectsp_6.html#K1">Carrier</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><a NAME="E1:14_1"/><i><font color="Green">E6</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:14_1"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="lmod_7.html#E1:14_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="vectsp_6.html#T19">VECTSP_6:19</a></i>;<br/>
<b>thus </b><a NAME="E5:14_1"/>
contradiction
 <b>by </b><i><a class="txt" href="lmod_7.html#E1:14"><i><font color="Green">E4</font></i></a>, <a class="txt" href="lmod_7.html#E4:14_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="lmod_6.html#T5">LMOD_6:5</a></i>;<br/>


</div><b>end;</b></div>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="mod_3.html#K1">Lin</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><a NAME="E1:14_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="mod_3.html#K1">Lin</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T3">RLVECT_1:3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:14_2"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="mod_3.html#T11">MOD_3:11</a></i>;<br/><a NAME="E4:14_2"/>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="lmod_7.html#E2:14"><i><font color="Green">E5</font></i></a></i>;<br/><a NAME="E5:14_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="lmod_7.html#E3:14_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="vectsp_6.html#T45">VECTSP_6:45</a></i>;<br/><b>hence </b><a NAME="E6:14_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="mod_3.html#K1">Lin</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T19">VECTSP_4:19</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:14"/>
 <a href="mod_3.html#K1">Lin</a> <font color="Maroon">c<sub>3</sub></font> is <a href="realset2.html#V3">trivial</a>
 <b>by </b><i><a class="ref" href="anproj_1.html#D8">ANPROJ_1:def 8</a></i>;<br/>


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