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

<b>set </b><font color="Maroon">lf</font> =  <a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font>;<br/>
<b>thus </b><a NAME="E1:108_2_1"/>
 <a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font> is <a href="bilinear.html#V2" target="_self">additiveSAF</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2_1_1"><b>proof </b></a><div class="add">

<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">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <font color="Maroon">f</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="bilinear.html#D13" target="_self">BILINEAR:def 13</a><br/>

<b>set </b><font color="Maroon">flf</font> =  <a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">A</font>;<br/>
<b>consider </b><font color="Maroon">w</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E2:108_2_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="vectsp_4.html#K3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23">VECTSP10:23</a></i>;<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">t</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="hahnban1.html#D11">HAHNBAN1:def 11</a><br/>

<b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">t</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">t</font></span>)</span>,<font color="Maroon">A</font>
<b>by </b><i/>
<br/>.= 
<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#K4">+</a> <font color="Maroon">t</font></span>)</span>,<font color="Maroon">w</font>
<b>by </b><i>, <a class="ref" href="bilinear.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<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#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">t</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"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">A</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">t</font>,<font color="Maroon">w</font></span>)</span>
<b>by </b><i>, <a class="ref" href="bilinear.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">A</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">t</font>,<font color="Maroon">A</font></span>)</span>
<b>by </b><i>, <a class="ref" href="bilinear.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <span class="p3">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">t</font>,<font color="Maroon">A</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <span class="p3">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <span class="p3">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>

<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">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <font color="Maroon">f</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="bilinear.html#D15" target="_self">BILINEAR:def 15</a><br/>

<b>set </b><font color="Maroon">flf</font> =  <a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">A</font>;<br/>
<b>consider </b><font color="Maroon">w</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E3:108_2_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="vectsp_4.html#K3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23">VECTSP10:23</a></i>;<br/>
<b>let </b><font color="Maroon">v</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="hahnban1.html#D12">HAHNBAN1:def 12</a><br/><b>let </b><font color="Maroon">r</font> be   <a href="vectsp_1.html#NM4">Scalar</a> of ;<br/>



<b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span>,<font color="Maroon">A</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span>,<font color="Maroon">w</font>
<b>by </b><i>, <a class="ref" href="bilinear.html#T1" target="_self">Th1</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="group_1.html#K1">*</a> <span class="p1">(<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>
<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"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">A</font></span>)</span>
<b>by </b><i>, <a class="ref" href="bilinear.html#T1" target="_self">Th1</a></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"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <span class="p3">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i/>
;<br/>


</div>
