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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">Submodule</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">W2</font> be  <a href="vectsp_2.html#V3">strict</a>  <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">V</font>, <font color="Maroon">W3</font> be  <a href="vectsp_2.html#V3">strict</a>  <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">V</font>;<br/>







<b>assume </b><a NAME="E1:43"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">W1</font> is    <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">W2</font>
 ;<br/>

<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W3</font></span>)</span> <a href="rmod_3.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W3</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W3</font></span>)</span> <a href="rmod_3.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">W3</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W3</font></span>)</span> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W3</font></span>)</span> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W2</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="rmod_3.html#K1" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">W3</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W3</font></span>)</span></span>)</span> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W2</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W3</font></span>)</span> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W2</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">W1</font> <a href="rmod_3.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">W3</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">W1</font> <a href="rmod_3.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W3</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W2</font></span>)</span> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W3</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">W2</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W3</font>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E3:43"/>
<font color="Maroon">W1</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W3</font> is    <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">W2</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W3</font>
 <b>by </b><i/>;<br/>


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