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

<span class="kw">set </span><font color="Maroon" title="c5">L</font> =  <a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">vq</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">Cv</font> =  <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> <font color="Maroon" title="c2">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">aCv</font> =  <a href="vectsp10.html#K3" title="VECTSP10:func.3">addCoset</a> <font color="Maroon" title="c2">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">mCv</font> =  <a href="vectsp10.html#K5" title="VECTSP10:func.5">lmultCoset</a> <font color="Maroon" title="c2">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">R</font> =  <a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">wq</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">Cw</font> =  <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> <font color="Maroon" title="c3">W</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">aCw</font> =  <a href="vectsp10.html#K3" title="VECTSP10:func.3">addCoset</a> <font color="Maroon" title="c3">W</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">mCw</font> =  <a href="vectsp10.html#K5" title="VECTSP10:func.5">lmultCoset</a> <font color="Maroon" title="c3">W</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">v</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><br/> for <font color="Olive" title="b2">w</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span> holds <br/>$3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>;<br/>
<div><span class="lab"><font color="Green" title="E45">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">A</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">B</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>  ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">A</font>,<font color="Olive" title="b1">B</font>,<font color="Olive" title="b2">y</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c16">B</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">A</font>,<font color="Maroon" title="c16">B</font>,<font color="Olive" title="b1">y</font>]</span><br/><span class="kw">consider </span><font color="Maroon" title="c17">a</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E2:110_1_1_1"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
<font color="Maroon" title="c15">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c18">b</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font><span class="kw"> such that </span><br/><a NAME="E4:110_1_1_1"/><span class="lab"><font color="Green" title="E47">A3</font></span>: 
<font color="Maroon" title="c16">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c19">y</font> = <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c18">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">A</font>,<font color="Maroon" title="c16">B</font>,<font color="Maroon" title="c19">y</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c20">a1</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">b1</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font>  st <font color="Maroon" title="c15">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a1</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; <font color="Maroon" title="c16">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b1</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span> holds <br/><font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">a1</font>,<font color="Olive" title="b1">b1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c21">b1</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a1</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; <font color="Maroon" title="c16">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">b1</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span> implies <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">a1</font>,<font color="Maroon" title="c21">b1</font> )</span><br/><span class="kw">assume </span><a NAME="E1:110_1_1_1_1"/>
<font color="Maroon" title="c15">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a1</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">b1</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span> implies <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">a1</font>,<font color="Maroon" title="c21">b1</font> )</span><br/><a NAME="E2:110_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">a1</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1_1_1"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="ref" href="vectsp_4.html#T59" title="VECTSP_4:th.59">VECTSP_4:59</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">vv</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E4:110_1_1_1_1"/><span class="lab"><font color="Green" title="E48">A4</font></span>: 
<font color="Maroon" title="c22">vv</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">and </span><br/><a NAME="E5:110_1_1_1_1"/><span class="lab"><font color="Green" title="E49">A5</font></span>: 
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c22">vv</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T57" title="VECTSP_4:th.57">VECTSP_4:57</a></span>;<br/><a NAME="E6:110_1_1_1_1"/>
<font color="Maroon" title="c22">vv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110_1_1_1_1"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E7:110_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">vv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#D19" target="_self" title="BILINEAR:def.19">Def19</a></span>;<br/><a NAME="E8:110_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A6</font></span>: 
 ex <font color="Olive" title="b1">aa</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Olive" title="b1">aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">vv</font> &amp; ( for <font color="Olive" title="b2">w0</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font> holds  <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">aa</font>,<font color="Olive" title="b2">w0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> ) )
 ;<br/><span class="kw">assume </span><a NAME="E9:110_1_1_1_1"/>
<font color="Maroon" title="c16">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">b1</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">a1</font>,<font color="Maroon" title="c21">b1</font></span><br/><a NAME="E10:110_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">b1</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110_1_1_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="vectsp_4.html#T59" title="VECTSP_4:th.59">VECTSP_4:59</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">ww</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font><span class="kw"> such that </span><br/><a NAME="E12:110_1_1_1_1"/><span class="lab"><font color="Green" title="E51">A7</font></span>: 
<font color="Maroon" title="c23">ww</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">and </span><br/><a NAME="E13:110_1_1_1_1"/><span class="lab"><font color="Green" title="E52">A8</font></span>: 
<font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c23">ww</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T57" title="VECTSP_4:th.57">VECTSP_4:57</a></span>;<br/><a NAME="E14:110_1_1_1_1"/>
<font color="Maroon" title="c23">ww</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:110_1_1_1_1"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E15:110_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">ww</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#D20" target="_self" title="BILINEAR:def.20">Def20</a></span>;<br/><a NAME="E16:110_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A9</font></span>: 
 ex <font color="Olive" title="b1">bb</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font> st <br/>( <font color="Olive" title="b1">bb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">ww</font> &amp; ( for <font color="Olive" title="b2">v0</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font> holds  <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">v0</font>,<font color="Olive" title="b1">bb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> ) )
 ;<br/><span class="kw">thus </span><font color="Maroon" title="c19">y</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">a1</font>,<font color="Maroon" title="c21">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">a1</font>,<font color="Maroon" title="c23">ww</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">vv</font>,<font color="Maroon" title="c21">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">vv</font>,<font color="Maroon" title="c23">ww</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1_1_1_1"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="txt" href="#E13:110_1_1_1_1"><span class="lab"><font color="Green" title="E52">A8</font></span></a>, <a class="ref" href="bilinear.html#T29" target="_self" title="BILINEAR:th.29">Th29</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">a1</font>,<font color="Maroon" title="c21">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">vv</font>,<font color="Maroon" title="c21">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">vv</font>,<font color="Maroon" title="c23">ww</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:110_1_1_1_1"><span class="lab"><font color="Green" title="E53">A9</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">a1</font>,<font color="Maroon" title="c21">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">vv</font>,<font color="Maroon" title="c23">ww</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:110_1_1_1_1"><span class="lab"><font color="Green" title="E50">A6</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">a1</font>,<font color="Maroon" title="c21">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">vv</font>,<font color="Maroon" title="c23">ww</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">a1</font>,<font color="Maroon" title="c21">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">vv</font>,<font color="Maroon" title="c23">ww</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">a1</font>,<font color="Maroon" title="c21">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:110_1_1_1_1"><span class="lab"><font color="Green" title="E50">A6</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">a1</font>,<font color="Maroon" title="c21">b1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:110_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">A</font>,<font color="Maroon" title="c16">B</font>,<font color="Maroon" title="c19">y</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c15">ff</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p3">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p3">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font><span class="kw"> such that </span><br/><a NAME="E3:110_1_1"/><span class="lab"><font color="Green" title="E46">A10</font></span>: 
for <font color="Olive" title="b1">A</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span><br/> for <font color="Olive" title="b2">B</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>,<font color="Maroon" title="c15">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="#E1:110_1_1"><span class="lab"><font color="Green" title="E45">A1</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c16">ff</font> = <font color="Maroon" title="c15">ff</font> as   <a href="bilinear.html#NM1" title="BILINEAR:NM.1">Form</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> ;<br/>
<a NAME="E5:110_1_1"/><span class="lab"><font color="Green" title="E47">A11</font></span>: 
 <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> <font color="Maroon" title="c2">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#D6" title="VECTSP10:def.6">VECTSP10:def 6</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E48">A12</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">ww</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font> is  <a href="hahnban1.html#V1" title="HAHNBAN1:attr.1">homogeneous</a> </span><br/><span class="kw">consider </span><font color="Maroon" title="c18">w</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font><span class="kw"> such that </span><br/><a NAME="E2:110_1_1_2"/><span class="lab"><font color="Green" title="E49">A13</font></span>: 
<font color="Maroon" title="c17">ww</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">w</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c19">ffw</font> =  <a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c20">A</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> holds  <span class="p1">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c20">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">A</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c21">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c20">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">A</font></span>)</span></span><br/><span class="kw">consider </span><font color="Maroon" title="c22">a</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E2:110_1_1_2_1"/><span class="lab"><font color="Green" title="E50">A14</font></span>: 
<font color="Maroon" title="c20">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">a</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></span>;<br/><a NAME="E3:110_1_1_2_1"/><span class="lab"><font color="Green" title="E51">A15</font></span>: 
( the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="vectsp10.html#K5" title="VECTSP10:func.5">lmultCoset</a> <font color="Maroon" title="c2">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="vectsp10.html#K5" title="VECTSP10:func.5">lmultCoset</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c20">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c22">a</font></span>)</span> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1_1"><span class="lab"><font color="Green" title="E47">A11</font></span></a>, <a class="txt" href="#E2:110_1_1_2_1"><span class="lab"><font color="Green" title="E50">A14</font></span></a>, <a class="ref" href="vectsp10.html#D5" title="VECTSP10:def.5">VECTSP10:def 5</a>, <a class="ref" href="vectsp10.html#D6" title="VECTSP10:def.6">VECTSP10:def 6</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c20">A</font></span>)</span> = 
<font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c20">A</font></span>)</span>,<font color="Maroon" title="c17">ww</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T10" target="_self" title="BILINEAR:th.10">Th10</a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c22">a</font></span>)</span>,<font color="Maroon" title="c18">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E2:110_1_1_2"><span class="lab"><font color="Green" title="E49">A13</font></span></a>, <a class="txt" href="#E3:110_1_1_2_1"><span class="lab"><font color="Green" title="E51">A15</font></span></a>, <a class="ref" href="vectsp_1.html#D24" title="VECTSP_1:def.24">VECTSP_1:def 24</a></span>
<br/>.= 
<font color="Maroon" title="c21">r</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c18">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T32" target="_self" title="BILINEAR:th.32">Th32</a></span>
<br/>.= 
<font color="Maroon" title="c21">r</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">A</font>,<font color="Maroon" title="c17">ww</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E2:110_1_1_2"><span class="lab"><font color="Green" title="E49">A13</font></span></a>, <a class="txt" href="#E2:110_1_1_2_1"><span class="lab"><font color="Green" title="E50">A14</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c21">r</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T10" target="_self" title="BILINEAR:th.10">Th10</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:110_1_1_2"/>
 <a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font> is  <a href="hahnban1.html#V1" title="HAHNBAN1:attr.1">homogeneous</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D12" title="HAHNBAN1:def.12">HAHNBAN1:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:110_1_1"/><span class="lab"><font color="Green" title="E49">A16</font></span>: 
 <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> <font color="Maroon" title="c3">W</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#D6" title="VECTSP10:def.6">VECTSP10:def 6</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E50">A17</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">vv</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font> is  <a href="hahnban1.html#V1" title="HAHNBAN1:attr.1">homogeneous</a> </span><br/><span class="kw">consider </span><font color="Maroon" title="c18">v</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E2:110_1_1_3"/><span class="lab"><font color="Green" title="E51">A18</font></span>: 
<font color="Maroon" title="c17">vv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c19">ffv</font> =  <a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c20">A</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> holds  <span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c20">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">A</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c21">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c20">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">A</font></span>)</span></span><br/><span class="kw">consider </span><font color="Maroon" title="c22">a</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font><span class="kw"> such that </span><br/><a NAME="E2:110_1_1_3_1"/><span class="lab"><font color="Green" title="E52">A19</font></span>: 
<font color="Maroon" title="c20">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">a</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></span>;<br/><a NAME="E3:110_1_1_3_1"/><span class="lab"><font color="Green" title="E53">A20</font></span>: 
( the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="vectsp10.html#K5" title="VECTSP10:func.5">lmultCoset</a> <font color="Maroon" title="c3">W</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="vectsp10.html#K5" title="VECTSP10:func.5">lmultCoset</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c20">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c22">a</font></span>)</span> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:110_1_1"><span class="lab"><font color="Green" title="E49">A16</font></span></a>, <a class="txt" href="#E2:110_1_1_3_1"><span class="lab"><font color="Green" title="E52">A19</font></span></a>, <a class="ref" href="vectsp10.html#D5" title="VECTSP10:def.5">VECTSP10:def 5</a>, <a class="ref" href="vectsp10.html#D6" title="VECTSP10:def.6">VECTSP10:def 6</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c20">A</font></span>)</span> = 
<font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c17">vv</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c20">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T9" target="_self" title="BILINEAR:th.9">Th9</a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c18">v</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c22">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E2:110_1_1_3"><span class="lab"><font color="Green" title="E51">A18</font></span></a>, <a class="txt" href="#E3:110_1_1_3_1"><span class="lab"><font color="Green" title="E53">A20</font></span></a>, <a class="ref" href="vectsp_1.html#D24" title="VECTSP_1:def.24">VECTSP_1:def 24</a></span>
<br/>.= 
<font color="Maroon" title="c21">r</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c18">v</font>,<font color="Maroon" title="c22">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T33" target="_self" title="BILINEAR:th.33">Th33</a></span>
<br/>.= 
<font color="Maroon" title="c21">r</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c17">vv</font>,<font color="Maroon" title="c20">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E2:110_1_1_3"><span class="lab"><font color="Green" title="E51">A18</font></span></a>, <a class="txt" href="#E2:110_1_1_3_1"><span class="lab"><font color="Green" title="E52">A19</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c21">r</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T9" target="_self" title="BILINEAR:th.9">Th9</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:110_1_1_3"/>
 <a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font> is  <a href="hahnban1.html#V1" title="HAHNBAN1:attr.1">homogeneous</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D12" title="HAHNBAN1:def.12">HAHNBAN1:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E51">A21</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">ww</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> </span><br/><span class="kw">consider </span><font color="Maroon" title="c18">w</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font><span class="kw"> such that </span><br/><a NAME="E2:110_1_1_4"/><span class="lab"><font color="Green" title="E52">A22</font></span>: 
<font color="Maroon" title="c17">ww</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">w</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c19">ffw</font> =  <a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c20">A</font>, <font color="Maroon" title="c21">B</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">A</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c21">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">A</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">B</font></span>)</span></span><br/><span class="kw">consider </span><font color="Maroon" title="c22">a</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E2:110_1_1_4_1"/><span class="lab"><font color="Green" title="E53">A23</font></span>: 
<font color="Maroon" title="c20">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">a</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c23">b</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E4:110_1_1_4_1"/><span class="lab"><font color="Green" title="E54">A24</font></span>: 
<font color="Maroon" title="c21">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">b</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></span>;<br/><a NAME="E5:110_1_1_4_1"/><span class="lab"><font color="Green" title="E55">A25</font></span>: 
( the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="vectsp10.html#K3" title="VECTSP10:func.3">addCoset</a> <font color="Maroon" title="c2">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="vectsp10.html#K3" title="VECTSP10:func.3">addCoset</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c20">A</font>,<font color="Maroon" title="c21">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c23">b</font></span>)</span> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1_1"><span class="lab"><font color="Green" title="E47">A11</font></span></a>, <a class="txt" href="#E2:110_1_1_4_1"><span class="lab"><font color="Green" title="E53">A23</font></span></a>, <a class="txt" href="#E4:110_1_1_4_1"><span class="lab"><font color="Green" title="E54">A24</font></span></a>, <a class="ref" href="vectsp10.html#D3" title="VECTSP10:def.3">VECTSP10:def 3</a>, <a class="ref" href="vectsp10.html#D6" title="VECTSP10:def.6">VECTSP10:def 6</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">A</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c21">B</font></span>)</span> = 
<font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">A</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c21">B</font></span>)</span>,<font color="Maroon" title="c17">ww</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T10" target="_self" title="BILINEAR:th.10">Th10</a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c23">b</font></span>)</span>,<font color="Maroon" title="c18">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E2:110_1_1_4"><span class="lab"><font color="Green" title="E52">A22</font></span></a>, <a class="txt" href="#E5:110_1_1_4_1"><span class="lab"><font color="Green" title="E55">A25</font></span></a>, <a class="ref" href="rlvect_1.html#T5" title="RLVECT_1:th.5">RLVECT_1:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c18">w</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c23">b</font>,<font color="Maroon" title="c18">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T27" target="_self" title="BILINEAR:th.27">Th27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">A</font>,<font color="Maroon" title="c17">ww</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c23">b</font>,<font color="Maroon" title="c18">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E2:110_1_1_4"><span class="lab"><font color="Green" title="E52">A22</font></span></a>, <a class="txt" href="#E2:110_1_1_4_1"><span class="lab"><font color="Green" title="E53">A23</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">A</font>,<font color="Maroon" title="c17">ww</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c21">B</font>,<font color="Maroon" title="c17">ww</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E2:110_1_1_4"><span class="lab"><font color="Green" title="E52">A22</font></span></a>, <a class="txt" href="#E4:110_1_1_4_1"><span class="lab"><font color="Green" title="E54">A24</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">A</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c21">B</font>,<font color="Maroon" title="c17">ww</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T10" target="_self" title="BILINEAR:th.10">Th10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">A</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T10" target="_self" title="BILINEAR:th.10">Th10</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:110_1_1_4"/>
 <a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">vv</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> </span><br/><span class="kw">consider </span><font color="Maroon" title="c18">v</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E2:110_1_1_5"/><span class="lab"><font color="Green" title="E52">A26</font></span>: 
<font color="Maroon" title="c17">vv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c19">ffv</font> =  <a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c20">A</font>, <font color="Maroon" title="c21">B</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">A</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c21">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">A</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">B</font></span>)</span></span><br/><span class="kw">consider </span><font color="Maroon" title="c22">a</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font><span class="kw"> such that </span><br/><a NAME="E2:110_1_1_5_1"/><span class="lab"><font color="Green" title="E53">A27</font></span>: 
<font color="Maroon" title="c20">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">a</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c23">b</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font><span class="kw"> such that </span><br/><a NAME="E4:110_1_1_5_1"/><span class="lab"><font color="Green" title="E54">A28</font></span>: 
<font color="Maroon" title="c21">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">b</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></span>;<br/><a NAME="E5:110_1_1_5_1"/><span class="lab"><font color="Green" title="E55">A29</font></span>: 
( the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="vectsp10.html#K3" title="VECTSP10:func.3">addCoset</a> <font color="Maroon" title="c3">W</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="vectsp10.html#K3" title="VECTSP10:func.3">addCoset</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c20">A</font>,<font color="Maroon" title="c21">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c23">b</font></span>)</span> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:110_1_1"><span class="lab"><font color="Green" title="E49">A16</font></span></a>, <a class="txt" href="#E2:110_1_1_5_1"><span class="lab"><font color="Green" title="E53">A27</font></span></a>, <a class="txt" href="#E4:110_1_1_5_1"><span class="lab"><font color="Green" title="E54">A28</font></span></a>, <a class="ref" href="vectsp10.html#D3" title="VECTSP10:def.3">VECTSP10:def 3</a>, <a class="ref" href="vectsp10.html#D6" title="VECTSP10:def.6">VECTSP10:def 6</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">A</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c21">B</font></span>)</span> = 
<font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c17">vv</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">A</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c21">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T9" target="_self" title="BILINEAR:th.9">Th9</a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c18">v</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c23">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E2:110_1_1_5"><span class="lab"><font color="Green" title="E52">A26</font></span></a>, <a class="txt" href="#E5:110_1_1_5_1"><span class="lab"><font color="Green" title="E55">A29</font></span></a>, <a class="ref" href="rlvect_1.html#T5" title="RLVECT_1:th.5">RLVECT_1:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c18">v</font>,<font color="Maroon" title="c22">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c18">v</font>,<font color="Maroon" title="c23">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T28" target="_self" title="BILINEAR:th.28">Th28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c17">vv</font>,<font color="Maroon" title="c20">A</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c18">v</font>,<font color="Maroon" title="c23">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E2:110_1_1_5"><span class="lab"><font color="Green" title="E52">A26</font></span></a>, <a class="txt" href="#E2:110_1_1_5_1"><span class="lab"><font color="Green" title="E53">A27</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c17">vv</font>,<font color="Maroon" title="c20">A</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c17">vv</font>,<font color="Maroon" title="c21">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E2:110_1_1_5"><span class="lab"><font color="Green" title="E52">A26</font></span></a>, <a class="txt" href="#E4:110_1_1_5_1"><span class="lab"><font color="Green" title="E54">A28</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">A</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c17">vv</font>,<font color="Maroon" title="c21">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T9" target="_self" title="BILINEAR:th.9">Th9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">A</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T9" target="_self" title="BILINEAR:th.9">Th9</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:110_1_1_5"/>
 <a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">ff</font> = <font color="Maroon" title="c16">ff</font> as   <a href="bilinear.html#NM2" title="BILINEAR:NM.2">bilinear-Form</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:110_1_1"><span class="lab"><font color="Green" title="E51">A21</font></span></a>, <a class="txt" href="#E6:110_1_1"><span class="lab"><font color="Green" title="E48">A12</font></span></a>, <a class="txt" href="#E8:110_1_1"><span class="lab"><font color="Green" title="E50">A17</font></span></a>, <a class="ref" href="bilinear.html#D12" target="_self" title="BILINEAR:def.12">Def12</a>, <a class="ref" href="bilinear.html#D13" target="_self" title="BILINEAR:def.13">Def13</a>, <a class="ref" href="bilinear.html#D14" target="_self" title="BILINEAR:def.14">Def14</a>, <a class="ref" href="bilinear.html#D15" target="_self" title="BILINEAR:def.15">Def15</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c17">ff</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span><br/> for <font color="Olive" title="b2">B</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span><br/> for <font color="Olive" title="b3">v</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><br/> for <font color="Olive" title="b4">w</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span> holds <br/><font color="Maroon" title="c17">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">w</font></span><br/>

<span class="kw">thus </span><a NAME="E12:110_1_1"/>
for <font color="Olive" title="b1">A</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span><br/> for <font color="Olive" title="b2">B</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span><br/> for <font color="Olive" title="b3">v</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><br/> for <font color="Olive" title="b4">w</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span> holds <br/><font color="Maroon" title="c17">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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