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<b>let </b><font color="Maroon">R</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_2.html#NM7">RightMod</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">W1</font> be    <a href="rmod_2.html#M1" target="_self">Submodule</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">W2</font> be    <a href="rmod_2.html#M1" target="_self">Submodule</a> of <font color="Maroon">V</font>;<br/>







<b>assume </b><a NAME="E1:31"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">v</font> being  <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>  st <font color="Olive">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> holds <br/><font color="Olive">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font>
 ;<br/>

<a NAME="E2:31"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>
 ;<br/>

<a NAME="E2:31_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">x</font> as   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/>
<a NAME="E4:31_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E5:31_1"/><b>then </b>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:31_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:31"/>
<font color="Maroon">W1</font> is    <a href="rmod_2.html#M1" target="_self">Submodule</a> of <font color="Maroon">W2</font>
 <b>by </b><i/>;<br/>


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