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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="vectsp_2.html#NM2">domRing</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="vectsp_2.html#NM7">RightMod</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><a NAME="E1:14"/><i><font color="Green">E9</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>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:14_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rmod_5.html#K1" target="_self">Lin</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="rmod_4.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:14_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="rmod_4.html#K5">Sum</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="rmod_5.html#T9" target="_self">Th9</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="rmod_4.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="rmod_5.html#E1:14"><i><font color="Green">E9</font></i></a>, <a class="ref" href="rmod_4.html#T25">RMOD_4:25</a></i>;<br/><a NAME="E5:14_1"/>
 <a href="rmod_4.html#K5">Sum</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="rmod_5.html#E3:14_1"><i><font color="Green">E10</font></i></a></i>;<br/><b>hence </b><a NAME="E6:14_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rmod_5.html#K1" target="_self">Lin</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="rmod_5.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:14"/>
 <a href="rmod_5.html#K1" target="_self">Lin</a> <font color="Maroon">c<sub>3</sub></font> is    <a href="rmod_2.html#M1">Submodule</a> of  <a href="rmod_5.html#K1" target="_self">Lin</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="rmod_2.html#T36">RMOD_2:36</a></i>;<br/>


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