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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#NM7">RightMod</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="rmod_2.html#M1">Submodule</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="vectsp_2.html#V3">strict</a>  <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>thus </b><a NAME="E1:11"/>
( <font color="Maroon">c<sub>3</sub></font> is    <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:11_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="rmod_2.html#D2">RMOD_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:11_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="rmod_3.html#E12"><i><font color="Green">Lemma7</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:11"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>3</sub></font> is    <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="rmod_3.html#T11" target="_self">Th11</a></i>;<br/>


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