<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">K</font> be  non <a href="struct_0.html#V3">empty</a> <a href="vectsp_1.html#V4">right-distributive</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">V</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of <font color="Maroon">K</font>, <font color="Maroon">W</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">r</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#NM1">Form</a> of <font color="Maroon">V</font>,<font color="Maroon">W</font>, <font color="Maroon">g</font> be   <a href="bilinear.html#NM1">Form</a> of <font color="Maroon">V</font>,<font color="Maroon">W</font>;<br/>







<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </b></a><div class="add"><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">w</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="bilinear.html#K3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="bilinear.html#K2" target="_self">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">w</font> = 
<font color="Maroon">r</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="bilinear.html#K2" target="_self">+</a> <font color="Maroon">g</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">w</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">r</font> <a href="group_1.html#K1">*</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> <font color="Maroon">v</font>,<font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">w</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="group_1.html#K1">*</a> <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">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">w</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D11">VECTSP_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="bilinear.html#K3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">w</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="bilinear.html#K3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="bilinear.html#K3" target="_self">*</a> <font color="Maroon">g</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">w</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="bilinear.html#K3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="bilinear.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="bilinear.html#K3" target="_self">*</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">w</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:14"/>
<font color="Maroon">r</font> <a href="bilinear.html#K3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="bilinear.html#K2" target="_self">+</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="bilinear.html#K3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="bilinear.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="bilinear.html#K3" target="_self">*</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


</div>
