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<b>let </b><font color="Maroon">GF</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">M</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">GF</font>;<br/><b>let </b><font color="Maroon">W1</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">W2</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">M</font>;<br/>







<b>thus </b><a NAME="E1:52"/>
( <font color="Maroon">W1</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W2</font> <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">M</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">M</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">M</font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">M</font> #) implies for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  ex <font color="Olive">v1</font>, <font color="Olive">v2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <br/>( <font color="Olive">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Olive">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">W1</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W2</font> <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">M</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">M</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">M</font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">M</font> #)
 ;<br/>

<b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>;<br/>

<a NAME="E2:52_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T3">RLVECT_1:3</a></i>;<br/>
<b>hence </b><a NAME="E3:52_1"/>
 ex <font color="Olive">v1</font>, <font color="Olive">v2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <br/>( <font color="Olive">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Olive">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v2</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:52"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  ex <font color="Olive">v1</font>, <font color="Olive">v2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <br/>( <font color="Olive">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Olive">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v2</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:52_2"/>
<font color="Maroon">W1</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W2</font> is    <a href="vectsp_4.html#M1">Subspace</a> of  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>;<br/><a NAME="E2:52_2"/>
 ex <font color="Olive">v1</font>, <font color="Olive">v2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <br/>( <font color="Olive">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Olive">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Olive">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v2</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:52_2"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:52"/>
<font color="Maroon">W1</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W2</font> <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">M</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">M</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">M</font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">M</font> #)
 <b>by </b><i><a class="ref" href="vectsp_4.html#T40">VECTSP_4:40</a></i>;<br/>


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