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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>assume </b><a NAME="E1:53"/>
<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="E2:53"/><b>then </b>
(  <a href="vectsp_2.html#G1">RightModStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="vectsp_2.html#U1">rmult</a> of <font color="Maroon">V</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">W1</font> <a href="rmod_3.html#K2" target="_self">/\</a> <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a>  <a href="rmod_2.html#K1">(0).</a> <font color="Maroon">V</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:53"/><b>then </b>
(  <a href="vectsp_2.html#G1">RightModStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="vectsp_2.html#U1">rmult</a> of <font color="Maroon">V</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">W2</font> <a href="rmod_3.html#K2" target="_self">/\</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a>  <a href="rmod_2.html#K1">(0).</a> <font color="Maroon">V</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:53"/>
<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/>;<br/>


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