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





<b>consider </b><font color="Maroon">W'</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E2:16"/><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 <span class="p1">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">M</font></span>)</span>
 ;<br/>
<a NAME="E3:16"/><i><font color="Green">E32</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W'</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/>
<a NAME="E4:16"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">W'</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/>
<font color="Maroon">W</font> <a href="vectsp_5.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">M</font></span>)</span> = 
<font color="Maroon">W</font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W'</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">W'</font>
<b>by </b><i>, </i>
<br/>.= 
 <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#T39">VECTSP_4:39</a></i>
;<br/>
<b>hence </b><a NAME="E6:16"/>
( <span class="p1">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">M</font></span>)</span> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">W</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> #) &amp; <font color="Maroon">W</font> <a href="vectsp_5.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">M</font></span>)</span> <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/>;<br/>


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