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<b>let </b><font color="Maroon">K</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">V</font> be   <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/>





<b>set </b><font color="Maroon">vw</font> =  <a href="vectsp10.html#K6" target="_self">VectQuot</a> <font color="Maroon">V</font>,<font color="Maroon">W</font>;<br/>
<b>set </b><font color="Maroon">lm</font> = the <a href="vectsp_1.html#U2">lmult</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" target="_self">VectQuot</a> <font color="Maroon">V</font>,<font color="Maroon">W</font></span>)</span>;<br/>
<b>let </b><font color="Maroon">A</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" target="_self">VectQuot</a> <font color="Maroon">V</font>,<font color="Maroon">W</font></span>)</span>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="vectsp_1.html#NM4">Scalar</a> of ;<br/>





<b>assume </b><a NAME="E1:42"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E2:42"/>
<font color="Maroon">A</font> is    <a href="vectsp_4.html#M2">Coset</a> of <font color="Maroon">W</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:42"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is    <a href="vectsp_4.html#M2">Coset</a> of <font color="Maroon">W</font> : verum </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">w</font> = <font color="Maroon">A</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">V</font>,<font color="Maroon">W</font> ;<br/>
<b>thus </b><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">A</font> = 
the <a href="vectsp_1.html#U2">lmult</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" target="_self">VectQuot</a> <font color="Maroon">V</font>,<font color="Maroon">W</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">V</font>,<font color="Maroon">W</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">w</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font>
<b>by </b><i>, </i>
;<br/>


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