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

<b>set </b><font color="Maroon">L</font> =  <a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">vq</font> =  <a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">R</font> =  <a href="bilinear.html#K16" target="_self">RKer</a> <font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">wq</font> =  <a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <font color="Maroon">f</font></span>)</span>;<br/>
<b>let </b><font color="Maroon">f1</font> be   <a href="bilinear.html#NM2">bilinear-Form</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<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>, <font color="Maroon">f2</font> be   <a href="bilinear.html#NM2">bilinear-Form</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<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>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:109_1_2"/><i><font color="Green">E171</font></i>: 
for <font color="Olive">A</font> being  <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">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span><br/> for <font color="Olive">B</font> being  <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><br/> for <font color="Olive">v</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><br/> for <font color="Olive">w</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font>  st <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> <a href="vectsp_4.html#K3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">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> holds <br/><font color="Maroon">f1</font> <a href="binop_1.html#K2">.</a> <font color="Olive">A</font>,<font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">v</font>,<font color="Olive">w</font>
 <b>and </b><br/><a NAME="E2:109_1_2"/><i><font color="Green">E172</font></i>: 
for <font color="Olive">A</font> being  <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">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span><br/> for <font color="Olive">B</font> being  <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><br/> for <font color="Olive">v</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><br/> for <font color="Olive">w</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font>  st <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> <a href="vectsp_4.html#K3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">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> holds <br/><font color="Maroon">f2</font> <a href="binop_1.html#K2">.</a> <font color="Olive">A</font>,<font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">v</font>,<font color="Olive">w</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_1"><b>now </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">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">B</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>;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E2:109_1_2_1"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="vectsp_4.html#K3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</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>consider </b><font color="Maroon">b</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E4:109_1_2_1"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</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>thus </b><font color="Maroon">f1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> = 
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i>, , <a class="txt" href="bilinear.html#E18"><i><font color="Green">Lemma49</font></i></a></i>
<br/>.= 
<font color="Maroon">f2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
<b>by </b><i>, , <a class="txt" href="bilinear.html#E18"><i><font color="Green">Lemma49</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:109_1_2"/>
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


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