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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>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:3"/><i><font color="Green">E2</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:3"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="lmod_5.html#V1" target="_self">linearly-independent</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lmod_5.html#D1" target="_self">LMOD_5:def 1</a><br/>

<b>assume </b><a NAME="E3:3"/><i><font color="Green">E4</font></i>: 
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="lmod_5.html#E1:3"><i><font color="Green">E2</font></i></a>, <a class="ref" href="vectsp_6.html#T25">VECTSP_6:25</a></i>;<br/>
<a NAME="E5:3"/>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="lmod_5.html#E2:3"><i><font color="Green">E3</font></i></a>, <a class="txt" href="lmod_5.html#E3:3"><i><font color="Green">E4</font></i></a>, <a class="ref" href="lmod_5.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E6:3"/>
 <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> 
 ;<br/>


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