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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="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">well-unital</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">distributive</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">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="E1:18"/><i><font color="Green" title="E6">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>, <font color="Maroon" title="c6">v1</font>, <font color="Maroon" title="c7">v2</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="E2:18"/>
<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"><font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 ;<br/>

<a NAME="E3:18"/><b>then </b>
( <span class="p1">(<span class="default"><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></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> &amp; <span class="p1">(<span class="default"><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></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v2</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E4:18"/>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">v2</font>
 <b>by </b><i><a class="txt" href="vectsp10.html#E1:18"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="vectsp_5.html#D6" title="VECTSP_5:def.6">VECTSP_5:def 6</a></i>;<br/>


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