<?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">C</font> =  <a href="vectsp10.html#K2">CosetSet</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">aC</font> =  <a href="vectsp10.html#K3">addCoset</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">mC</font> =  <a href="vectsp10.html#K5">lmultCoset</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/>
<a NAME="E1:106_1_1"/><i><font color="Green">E33</font></i>: 
 <a href="vectsp10.html#K2">CosetSet</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> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</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>
 
<b>by </b><i><a class="ref" href="vectsp10.html#D6">VECTSP10:def 6</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">a</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">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> holds <br/><font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Maroon">a<sub>2</sub></font>;<br/>
<div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1_1_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">w0</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font>;<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="E2:106_1_1_1"/><i><font color="Green">E38</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>take </b><font color="Maroon">y</font> = <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">w0</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a1</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><b>assume </b><a NAME="E1:106_1_1_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</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>
 ;<br/><a NAME="E2:106_1_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a1</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="vectsp_4.html#T59">VECTSP_4:59</a></i>;<br/><b>then consider </b><font color="Maroon">w</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:106_1_1_1_1"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">w</font> <a href="rlvect_1.html#R1">in</a>  <a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font> )
 <b>by </b><i><a class="ref" href="vectsp_4.html#T57">VECTSP_4:57</a></i>;<br/><a NAME="E5:106_1_1_1_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <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="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E6:106_1_1_1_1"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="bilinear.html#D8" target="_self">Def8</a></i>;<br/><b>then consider </b><font color="Maroon">aa</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E8:106_1_1_1_1"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">aa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> &amp; ( for <font color="Olive">ww</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font> holds  <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">aa</font>,<font color="Olive">ww</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> ) )
 ;<br/><b>thus </b><font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<font color="Maroon">w0</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">w</font>,<font color="Maroon">w0</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<font color="Maroon">w0</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<font color="Maroon">w0</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:106_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font>,<font color="Maroon">w0</font>,<font color="Maroon">y</font>]
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">ff</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p3">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font><b> such that </b><br/><a NAME="E4:106_1_1"/><i><font color="Green">E106</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">w</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>,<font color="Olive">w</font>,<font color="Maroon">ff</font> <a href="binop_1.html#K2">.</a> <font color="Olive">A</font>,<font color="Olive">w</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>(<a class="ref" href="bilinear.html#T1" target="_self">Th1</a>);<br/></i>
<b>reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">ff</font> as   <a href="bilinear.html#NM1">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>,<font color="Maroon">W</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1_1_2"><b>now </b></a><div class="add"><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>set </b><font color="Maroon">ffw</font> =  <a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">ff</font>,<font color="Maroon">w</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1_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>, <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">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>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="E2:106_1_1_2_1"/><i><font color="Green">E107</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">V</font><b> such that </b><br/><a NAME="E4:106_1_1_2_1"/><i><font color="Green">E108</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#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/><a NAME="E5:106_1_1_2_1"/><i><font color="Green">E109</font></i>: 
the <a href="rlvect_1.html#U1">add</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> <a href="funct_2.html#R1">=</a>  <a href="vectsp10.html#K3">addCoset</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>
 <b>by </b><i><a class="ref" href="vectsp10.html#D6">VECTSP10:def 6</a></i>;<br/><a NAME="E6:106_1_1_2_1"/><i><font color="Green">E110</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp10.html#K3">addCoset</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> <a href="binop_1.html#K1">.</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span> <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="txt" href="bilinear.html#E9"><i><font color="Green">Lemma44</font></i></a>, , , <a class="ref" href="vectsp10.html#D3">VECTSP10:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">ff</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">B</font></span>)</span> = 
<font color="Maroon">ff</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">w</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">ff</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p3">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">w</font>
<b>by </b><i>, <a class="txt" href="bilinear.html#E18"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="bilinear.html#T9" target="_self">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</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">b</font>,<font color="Maroon">w</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">A</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">b</font>,<font color="Maroon">w</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">A</font>,<font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">B</font>,<font color="Maroon">w</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">ff</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">B</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#K10" target="_self">FunctionalSAF</a> <font color="Maroon">ff</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <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#K10" target="_self">FunctionalSAF</a> <font color="Maroon">ff</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i/>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:106_1_1_2"/>
 <a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">ff</font>,<font color="Maroon">w</font> is <a href="hahnban1.html#V1">additive</a>
 <b>by </b><i><a class="ref" href="hahnban1.html#D11">HAHNBAN1:def 11</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">ff</font> as  <a href="bilinear.html#V2" target="_self">additiveSAF</a> <a href="bilinear.html#NM1">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>,<font color="Maroon">W</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1_1_3"><b>now </b></a><div class="add"><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>set </b><font color="Maroon">ffw</font> =  <a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">ff</font>,<font color="Maroon">w</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1_1_3_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">r</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>;<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="E2:106_1_1_3_1"/><i><font color="Green">E131</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/><a NAME="E3:106_1_1_3_1"/><i><font color="Green">E132</font></i>: 
the <a href="vectsp_1.html#U2">lmult</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> <a href="funct_2.html#R1">=</a>  <a href="vectsp10.html#K5">lmultCoset</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>
 <b>by </b><i><a class="ref" href="vectsp10.html#D6">VECTSP10:def 6</a></i>;<br/><a NAME="E4:106_1_1_3_1"/><i><font color="Green">E133</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp10.html#K5">lmultCoset</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> <a href="binop_1.html#K1">.</a> <font color="Maroon">r</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">a</font></span>)</span> <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="txt" href="bilinear.html#E9"><i><font color="Green">Lemma44</font></i></a>, , <a class="ref" href="vectsp10.html#D5">VECTSP10:def 5</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">ff</font>,<font color="Maroon">w</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">A</font></span>)</span> = 
<font color="Maroon">ff</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">A</font></span>)</span>,<font color="Maroon">w</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">ff</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="vectsp_1.html#U2">lmult</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p3">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">r</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></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">a</font></span>)</span>,<font color="Maroon">w</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">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">a</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"><font color="Maroon">ff</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">A</font>,<font color="Maroon">w</font></span>)</span>
<b>by </b><i>, </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> <font color="Maroon">ff</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i/>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:106_1_1_3"/>
 <a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">ff</font>,<font color="Maroon">w</font> is <a href="hahnban1.html#V2">homogeneous</a>
 <b>by </b><i><a class="ref" href="hahnban1.html#D12">HAHNBAN1:def 12</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">ff</font> as  <a href="bilinear.html#V2" target="_self">additiveSAF</a> <a href="bilinear.html#V4" target="_self">homogeneousSAF</a> <a href="bilinear.html#NM1">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>,<font color="Maroon">W</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">ff</font>
;<br/>

<b>thus </b><a NAME="E10:106_1_1"/>
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">w</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font><br/> for <font color="Olive">v</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</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> holds <br/><font color="Maroon">ff</font> <a href="binop_1.html#K2">.</a> <font color="Olive">A</font>,<font color="Olive">w</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>by </b><i/>;<br/>


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