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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="rmod_2.html#M1">Submodule</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/>









<b>assume </b><a NAME="E1:60"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">V</font> <a href="rmod_3.html#R1" target="_self">is_the_direct_sum_of</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font>
 ;<br/>

<a NAME="E2:60"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rmod_3.html#K4" target="_self">|--</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font></span>)</span> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rmod_3.html#K4" target="_self">|--</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font></span>)</span> <a href="domain_1.html#K3">`2</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rmod_3.html#K4" target="_self">|--</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font></span>)</span> <a href="domain_1.html#K2">`1</a>  <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rmod_3.html#K4" target="_self">|--</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font></span>)</span> <a href="domain_1.html#K3">`2</a>  <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:60"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">V</font> <a href="rmod_3.html#R1" target="_self">is_the_direct_sum_of</a> <font color="Maroon">W2</font>,<font color="Maroon">W1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:60"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rmod_3.html#K4" target="_self">|--</a> <font color="Maroon">W2</font>,<font color="Maroon">W1</font></span>)</span> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rmod_3.html#K4" target="_self">|--</a> <font color="Maroon">W2</font>,<font color="Maroon">W1</font></span>)</span> <a href="domain_1.html#K2">`1</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:60"/>
( <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rmod_3.html#K4" target="_self">|--</a> <font color="Maroon">W2</font>,<font color="Maroon">W1</font></span>)</span> <a href="domain_1.html#K2">`1</a>  <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rmod_3.html#K4" target="_self">|--</a> <font color="Maroon">W2</font>,<font color="Maroon">W1</font></span>)</span> <a href="domain_1.html#K3">`2</a>  <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> )
 
<b>by </b><i><a class="ref" href="rmod_3.html#T7" target="_self">Th7</a>, </i>;<br/>
<b>hence </b><a NAME="E6:60"/>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rmod_3.html#K4" target="_self">|--</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font></span>)</span> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rmod_3.html#K4" target="_self">|--</a> <font color="Maroon">W2</font>,<font color="Maroon">W1</font></span>)</span> <a href="domain_1.html#K3">`2</a> 
 <b>by </b><i>, <a class="ref" href="rmod_3.html#T5" target="_self">Th5</a>, <a class="txt" href="rmod_3.html#E10"><i><font color="Green">Lemma45</font></i></a>, </i>;<br/>


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