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<b>let </b><font color="Maroon" title="c1">K</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font>;<br/><b>let </b><font color="Maroon" title="c3">W1</font>, <font color="Maroon" title="c4">W2</font> be    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>;<br/>





<b>assume </b><a NAME="E1:22"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon" title="c2">V</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">v</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font>;<br/>

<b>assume </b><a NAME="E2:22"/>
<font color="Maroon" title="c5">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font>
 ;<br/>

<a NAME="E3:22"/><b>then </b>
<font color="Maroon" title="c5">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> <font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">v</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="vectsp10.html#E1:22"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="vectsp10.html#T10" target="_self" title="VECTSP10:th.10">Th10</a>, <a class="ref" href="vectsp_5.html#T51" title="VECTSP_5:th.51">VECTSP_5:51</a></i>;<br/>
<b>hence </b><a NAME="E4:22"/>
<font color="Maroon" title="c5">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span>,<font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="vectsp10.html#E1:22"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="vectsp10.html#T9" target="_self" title="VECTSP10:th.9">Th9</a>, <a class="ref" href="vectsp_5.html#T51" title="VECTSP_5:th.51">VECTSP_5:51</a></i>;<br/>


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