<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V4">associative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="vectsp_1.html#V12">VectSp-like</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:59"/>
( <span class="p1">(<span class="default"><a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="vectsp_5.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">c<sub>2</sub></font> #) &amp;  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="vectsp_5.html#K2" target="_self">/\</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T14" target="_self">Th14</a>, <a class="ref" href="vectsp_5.html#T25" target="_self">Th25</a></i>;<br/>
<b>hence </b><a NAME="E2:59"/>
<font color="Maroon">c<sub>2</sub></font> <a href="vectsp_5.html#R1" target="_self">is_the_direct_sum_of</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></font>, <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D4" target="_self">Def4</a></i>;<br/>

<b>hence </b><a NAME="E3:59"/>
<font color="Maroon">c<sub>2</sub></font> <a href="vectsp_5.html#R1" target="_self">is_the_direct_sum_of</a>  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">c<sub>2</sub></font>, <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E66"><i><font color="Green">Lemma40</font></i></a></i>;<br/>


</div>
