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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> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</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#M1" target="_self">Subspace</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>











<b>assume </b><a NAME="E1:16"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="vectsp_4.html#T18" target="_self">Th18</a></i>;<br/>
<i><font color="Green">E13</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>4</sub></font> = 
the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>

<br/>.= 
the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="vectsp_4.html#E1:16"><i><font color="Green">E12</font></i></a></i>
;<br/>
<font color="Maroon">c<sub>3</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>6</sub></font> = 
the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>

<br/>.= 
the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default">the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="vectsp_4.html#D2" target="_self">Def2</a></i>
;<br/>
<b>hence </b><a NAME="E5:16"/>
<font color="Maroon">c<sub>3</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_4.html#E3:16"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


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