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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></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">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1" target="_self">Subspace</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="vectsp_4.html#M2" target="_self">Coset</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="vectsp_4.html#M2" target="_self">Coset</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>









<a NAME="E1:75"/><i><font color="Green">E51</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="vectsp_4.html#K3" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E2:75"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="vectsp_4.html#K3" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#D6" target="_self">Def6</a></i>;<br/>
<b>hence </b><a NAME="E3:75"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="vectsp_4.html#E1:75"><i><font color="Green">E51</font></i></a>, <a class="ref" href="vectsp_4.html#T83" target="_self">Th83</a></i>;<br/>


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