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<div class="add">

<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">W</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">M</font>, <font color="Maroon">W'</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">M</font>, <font color="Maroon">W1</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">M</font>;<br/>





<b>assume </b><a NAME="E1:14"/><i><font color="Green">E29</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W'</font>
 ;<br/>

<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">W1W</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v2</font></span>)</span> where <font color="Olive">v1</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>, <font color="Olive">v2</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> : ( <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">W</font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon">W1W'</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v2</font></span>)</span> where <font color="Olive">v1</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>, <font color="Olive">v2</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> : ( <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">W'</font> ) </span>}</span> ;<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/><b>thus </b><a NAME="E1:14_1"/>
( <font color="Maroon">v</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">W</font> implies <font color="Maroon">v</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">W'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_1"/>
<font color="Maroon">v</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">W</font>
 ;<br/>

<a NAME="E2:14_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E3:14_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v2</font></span>)</span> where <font color="Olive">v1</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>, <font color="Olive">v2</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> : ( <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">W</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">v1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>, <font color="Maroon">v2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E5:14_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font>
 <b>and </b><br/><a NAME="E6:14_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E7:14_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 ;<br/>
<a NAME="E8:14_1_1"/>
<font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W'</font>
 
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E9:14_1_1"/><b>then </b>
<font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W'</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E10:14_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v2</font></span>)</span> where <font color="Olive">v1</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>, <font color="Olive">v2</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> : ( <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">W'</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:14_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W'</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E12:14_1_1"/>
<font color="Maroon">v</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">W'</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:14_1"/>
<font color="Maroon">v</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">W'</font>
 ;<br/><a NAME="E3:14_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W'</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E4:14_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v2</font></span>)</span> where <font color="Olive">v1</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>, <font color="Olive">v2</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> : ( <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">W'</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">v1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>, <font color="Maroon">v2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E6:14_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font>
 <b>and </b><br/><a NAME="E7:14_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E8:14_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W'</font>
 ;<br/><a NAME="E9:14_1"/>
<font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>
 <b>by </b><i>, , <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E10:14_1"/><b>then </b>
<font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E11:14_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v2</font></span>)</span> where <font color="Olive">v1</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>, <font color="Olive">v2</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> : ( <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">W</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><a NAME="E12:14_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E13:14_1"/>
<font color="Maroon">v</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">W</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:14"/>
<font color="Maroon">W1</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W'</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T38">VECTSP_4:38</a></i>;<br/>

<a NAME="E4:14"/>
( <font color="Maroon">W1</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">W1</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W'</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W1</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:14"/>
<font color="Maroon">W</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W'</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W1</font>
 <b>by </b><i>, <a class="ref" href="vectsp_4.html#T38">VECTSP_4:38</a></i>;<br/>


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