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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="E1:32"/>
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#K2" target="_self">/\</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W2</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:32_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x</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#K2" target="_self">/\</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W2</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:32_1"/>
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#K2" target="_self">/\</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="xboole_0.html#K3">/\</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">W2</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:32_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E1:32_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


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

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:32"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>
 ;<br/>

<a NAME="E3:32"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/>
<a NAME="E5:32"/>
( <font color="Maroon">x1</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp;  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">M</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> )
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a>, <a class="ref" href="vectsp_4.html#T25">VECTSP_4:25</a></i>;<br/>
<a NAME="E6:32"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>, <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> : ( <font color="Olive">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Olive">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> ) </span>}</span> 
 
;<br/>
<a NAME="E7:32"/><b>then </b>
<font color="Maroon">x</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">W2</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:32"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="xboole_0.html#K3">/\</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">W2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E9:32"/>
<font color="Maroon">x</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#K2" target="_self">/\</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W2</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


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