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

<b>set </b><font color="Maroon" title="c5">lf</font> =  <a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font>;<br/>
<b>thus </b><a NAME="E1:108_1_1"/>
 <a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font> is <a href="bilinear.html#V1" title="BILINEAR:attr.1">additiveFAF</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">A</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">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>; <i><font color="Red">:: according to </font></i><a class="ref" href="bilinear.html#D12" target="_self" title="BILINEAR:def.12">BILINEAR:def 12</a><br/>

<b>set </b><font color="Maroon" title="c7">flf</font> =  <a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</font>;<br/>
<b>consider </b><font color="Maroon" title="c8">v</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E2:108_1_1_1"/><i><font color="Green" title="E46">A1</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/>
<b>let </b><font color="Maroon" title="c9">w</font>, <font color="Maroon" title="c10">t</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c3">W</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="hahnban1.html#D11" title="HAHNBAN1:def.11">HAHNBAN1:def 11</a><br/>

<b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</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="c9">w</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c10">t</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">w</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c10">t</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T9" target="_self" title="BILINEAR:th.9">Th9</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">v</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">w</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c10">t</font></span>)</span>
<b>by </b><i><a class="txt" href="bilinear.html#E2:108_1_1_1"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="bilinear.html#D21" target="_self" title="BILINEAR:def.21">Def21</a></i>
<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="c8">v</font>,<font color="Maroon" title="c9">w</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</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="c8">v</font>,<font color="Maroon" title="c10">t</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T28" target="_self" title="BILINEAR:th.28">Th28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c9">w</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</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="c8">v</font>,<font color="Maroon" title="c10">t</font></span>)</span>
<b>by </b><i><a class="txt" href="bilinear.html#E2:108_1_1_1"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="bilinear.html#D21" target="_self" title="BILINEAR:def.21">Def21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c9">w</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c10">t</font></span>)</span>
<b>by </b><i><a class="txt" href="bilinear.html#E2:108_1_1_1"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="bilinear.html#D21" target="_self" title="BILINEAR:def.21">Def21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p3">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">w</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c10">t</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T9" target="_self" title="BILINEAR:th.9">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p3">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">w</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p3">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">t</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T9" target="_self" title="BILINEAR:th.9">Th9</a></i>
;<br/>


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

<b>let </b><font color="Maroon" title="c6">A</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">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>; <i><font color="Red">:: according to </font></i><a class="ref" href="bilinear.html#D14" target="_self" title="BILINEAR:def.14">BILINEAR:def 14</a><br/>

<b>set </b><font color="Maroon" title="c7">flf</font> =  <a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</font>;<br/>
<b>consider </b><font color="Maroon" title="c8">v</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E3:108_1_1"/><i><font color="Green" title="E46">A2</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/>
<b>let </b><font color="Maroon" title="c9">w</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c3">W</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="hahnban1.html#D12" title="HAHNBAN1:def.12">HAHNBAN1:def 12</a><br/><b>let </b><font color="Maroon" title="c10">r</font> be   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of ;<br/>



<b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</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="c10">r</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c9">w</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c9">w</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T9" target="_self" title="BILINEAR:th.9">Th9</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">v</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c9">w</font></span>)</span>
<b>by </b><i><a class="txt" href="bilinear.html#E3:108_1_1"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="bilinear.html#D21" target="_self" title="BILINEAR:def.21">Def21</a></i>
<br/>.= 
<font color="Maroon" title="c10">r</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</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="c8">v</font>,<font color="Maroon" title="c9">w</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T33" target="_self" title="BILINEAR:th.33">Th33</a></i>
<br/>.= 
<font color="Maroon" title="c10">r</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c9">w</font></span>)</span>
<b>by </b><i><a class="txt" href="bilinear.html#E3:108_1_1"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="bilinear.html#D21" target="_self" title="BILINEAR:def.21">Def21</a></i>
<br/>.= 
<font color="Maroon" title="c10">r</font> <a href="group_1.html#K1" title="GROUP_1:func.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> <span class="p3">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">w</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T9" target="_self" title="BILINEAR:th.9">Th9</a></i>
;<br/>


</div>
