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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>, <font color="Maroon">W</font> be   <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Maroon">w</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font>, <font color="Maroon">z</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="bilinear.html#NM2">bilinear-Form</a> of <font color="Maroon">V</font>,<font color="Maroon">W</font>;<br/>











<b>set </b><font color="Maroon">v1</font> = <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">y</font>;<br/>
<b>set </b><font color="Maroon">v3</font> = <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">z</font>;<br/>
<b>set </b><font color="Maroon">v4</font> = <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>;<br/>
<b>set </b><font color="Maroon">v5</font> = <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">w</font>,<font color="Maroon">z</font>;<br/>
<b>thus </b><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">w</font>,<span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">w</font>,<font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>


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