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

<b>set </b><font color="Maroon" title="c3">vq</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c2">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c4">vr</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>;<br/>
<a NAME="E1:125_1_1"/>
 <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="hermitan.html#T59" target="_self" title="HERMITAN:th.59">Th59</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">sc</font> =  <a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c2">f</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="c1">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c2">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="c1">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> ;<br/>
<a NAME="E3:125_1_1"/><i><font color="Green" title="E60">A1</font></i>: 
<font color="Maroon" title="c5">sc</font> is <a href="bilinear.html#V4" title="BILINEAR:attr.4">homogeneousSAF</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">w</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="c1">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="bilinear.html#D15" title="BILINEAR:def.15">BILINEAR:def 15</a><br/>

<b>reconsider </b><font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c6">w</font> as   <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="c1">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="hermitan.html#T59" target="_self" title="HERMITAN:th.59">Th59</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">wa</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:125_1_1_1"/><i><font color="Green" title="E61">A2</font></i>: 
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">wa</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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">fg</font> =  <a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c5">sc</font>,<font color="Maroon" title="c6">w</font>;<br/>
<b>let </b><font color="Maroon" title="c10">v</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="c1">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>; <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="c11">r</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Scalar</a> of ;<br/>



<b>consider </b><font color="Maroon" title="c12">va</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E5:125_1_1_1"/><i><font color="Green" title="E62">A3</font></i>: 
<font color="Maroon" title="c10">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">va</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="c2">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/>
<a NAME="E6:125_1_1_1"/><i><font color="Green" title="E63">A4</font></i>: 
<font color="Maroon" title="c11">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c12">va</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="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="hermitan.html#E5:125_1_1_1"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="vectsp10.html#T26" title="VECTSP10:th.26">VECTSP10:26</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c5">sc</font>,<font color="Maroon" title="c6">w</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="c11">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">v</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">v</font></span>)</span>,<font color="Maroon" title="c6">w</font>
<b>by </b><i><a class="ref" href="bilinear.html#T10" title="BILINEAR:th.10">BILINEAR:10</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c12">va</font></span>)</span>,<font color="Maroon" title="c8">wa</font>
<b>by </b><i><a class="txt" href="hermitan.html#E3:125_1_1_1"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="hermitan.html#E6:125_1_1_1"><i><font color="Green" title="E63">A4</font></i></a>, <a class="ref" href="hermitan.html#D12" target="_self" title="HERMITAN:def.12">Def12</a></i>
<br/>.= 
<font color="Maroon" title="c11">r</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">va</font>,<font color="Maroon" title="c8">wa</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T32" title="BILINEAR:th.32">BILINEAR:32</a></i>
<br/>.= 
<font color="Maroon" title="c11">r</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">sc</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c6">w</font></span>)</span>
<b>by </b><i><a class="txt" href="hermitan.html#E3:125_1_1_1"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="hermitan.html#E5:125_1_1_1"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="hermitan.html#D12" target="_self" title="HERMITAN:def.12">Def12</a></i>
<br/>.= 
<font color="Maroon" title="c11">r</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</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="c5">sc</font>,<font color="Maroon" title="c6">w</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">v</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T10" title="BILINEAR:th.10">BILINEAR:10</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:125_1_1"/><i><font color="Green" title="E61">A5</font></i>: 
<font color="Maroon" title="c5">sc</font> is <a href="bilinear.html#V2" title="BILINEAR:attr.2">additiveSAF</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">w</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="c1">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="bilinear.html#D13" title="BILINEAR:def.13">BILINEAR:def 13</a><br/>

<b>reconsider </b><font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c6">w</font> as   <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="c1">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="hermitan.html#T59" target="_self" title="HERMITAN:th.59">Th59</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">wa</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:125_1_1_2"/><i><font color="Green" title="E62">A6</font></i>: 
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">wa</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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">fg</font> =  <a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c5">sc</font>,<font color="Maroon" title="c6">w</font>;<br/>
<b>let </b><font color="Maroon" title="c10">v1</font>, <font color="Maroon" title="c11">v2</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="c1">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>; <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>consider </b><font color="Maroon" title="c12">va</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E5:125_1_1_2"/><i><font color="Green" title="E63">A7</font></i>: 
<font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">va</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="c2">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>consider </b><font color="Maroon" title="c13">vb</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E7:125_1_1_2"/><i><font color="Green" title="E64">A8</font></i>: 
<font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">vb</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="c2">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/>
<a NAME="E8:125_1_1_2"/><i><font color="Green" title="E65">A9</font></i>: 
<font color="Maroon" title="c10">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">va</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c13">vb</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="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="hermitan.html#E5:125_1_1_2"><i><font color="Green" title="E63">A7</font></i></a>, <a class="txt" href="hermitan.html#E7:125_1_1_2"><i><font color="Green" title="E64">A8</font></i></a>, <a class="ref" href="vectsp10.html#T27" title="VECTSP10:th.27">VECTSP10:27</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c5">sc</font>,<font color="Maroon" title="c6">w</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">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">v2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">v2</font></span>)</span>,<font color="Maroon" title="c6">w</font>
<b>by </b><i><a class="ref" href="bilinear.html#T10" title="BILINEAR:th.10">BILINEAR:10</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">va</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c13">vb</font></span>)</span>,<font color="Maroon" title="c8">wa</font>
<b>by </b><i><a class="txt" href="hermitan.html#E3:125_1_1_2"><i><font color="Green" title="E62">A6</font></i></a>, <a class="txt" href="hermitan.html#E8:125_1_1_2"><i><font color="Green" title="E65">A9</font></i></a>, <a class="ref" href="hermitan.html#D12" target="_self" title="HERMITAN:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">va</font>,<font color="Maroon" title="c8">wa</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="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">vb</font>,<font color="Maroon" title="c8">wa</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T27" title="BILINEAR:th.27">BILINEAR:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">sc</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c6">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="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">vb</font>,<font color="Maroon" title="c8">wa</font></span>)</span>
<b>by </b><i><a class="txt" href="hermitan.html#E3:125_1_1_2"><i><font color="Green" title="E62">A6</font></i></a>, <a class="txt" href="hermitan.html#E5:125_1_1_2"><i><font color="Green" title="E63">A7</font></i></a>, <a class="ref" href="hermitan.html#D12" target="_self" title="HERMITAN:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">sc</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c6">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="c5">sc</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c6">w</font></span>)</span>
<b>by </b><i><a class="txt" href="hermitan.html#E3:125_1_1_2"><i><font color="Green" title="E62">A6</font></i></a>, <a class="txt" href="hermitan.html#E7:125_1_1_2"><i><font color="Green" title="E64">A8</font></i></a>, <a class="ref" href="hermitan.html#D12" target="_self" title="HERMITAN:def.12">Def12</a></i>
<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="c5">sc</font>,<font color="Maroon" title="c6">w</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">v1</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="c5">sc</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c6">w</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T10" title="BILINEAR:th.10">BILINEAR:10</a></i>
<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="c5">sc</font>,<font color="Maroon" title="c6">w</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">v1</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="c5">sc</font>,<font color="Maroon" title="c6">w</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T10" title="BILINEAR:th.10">BILINEAR:10</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:125_1_1"/><i><font color="Green" title="E62">A10</font></i>: 
<font color="Maroon" title="c5">sc</font> is <a href="hermitan.html#V3" title="HERMITAN:attr.3">hermitan</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">v</font>, <font color="Maroon" title="c7">w</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="c1">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="hermitan.html#D5" target="_self" title="HERMITAN:def.5">HERMITAN:def 5</a><br/>

<b>reconsider </b><font color="Maroon" title="c8">A</font> = <font color="Maroon" title="c7">w</font> as   <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="c1">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="hermitan.html#T59" target="_self" title="HERMITAN:th.59">Th59</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">wa</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:125_1_1_3"/><i><font color="Green" title="E63">A11</font></i>: 
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">wa</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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">B</font> = <font color="Maroon" title="c6">v</font> as   <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="c1">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="hermitan.html#T59" target="_self" title="HERMITAN:th.59">Th59</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">va</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E6:125_1_1_3"/><i><font color="Green" title="E64">A12</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">va</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="c2">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/>
<a NAME="E7:125_1_1_3"/><i><font color="Green" title="E65">A13</font></i>: 
<font color="Maroon" title="c10">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">va</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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="hermitan.html#E6:125_1_1_3"><i><font color="Green" title="E64">A12</font></i></a>, <a class="ref" href="hermitan.html#T59" target="_self" title="HERMITAN:th.59">Th59</a></i>;<br/>
<a NAME="E8:125_1_1_3"/><i><font color="Green" title="E66">A14</font></i>: 
<font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">wa</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="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="hermitan.html#E3:125_1_1_3"><i><font color="Green" title="E63">A11</font></i></a>, <a class="ref" href="hermitan.html#T59" target="_self" title="HERMITAN:th.59">Th59</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c5">sc</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c7">w</font> = 
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">va</font>,<font color="Maroon" title="c9">wa</font>
<b>by </b><i><a class="txt" href="hermitan.html#E3:125_1_1_3"><i><font color="Green" title="E63">A11</font></i></a>, <a class="txt" href="hermitan.html#E6:125_1_1_3"><i><font color="Green" title="E64">A12</font></i></a>, <a class="ref" href="hermitan.html#D12" target="_self" title="HERMITAN:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">wa</font>,<font color="Maroon" title="c11">va</font></span>)</span> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> 
<b>by </b><i><a class="ref" href="hermitan.html#D5" target="_self" title="HERMITAN:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">sc</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">w</font>,<font color="Maroon" title="c6">v</font></span>)</span> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> 
<b>by </b><i><a class="txt" href="hermitan.html#E7:125_1_1_3"><i><font color="Green" title="E65">A13</font></i></a>, <a class="txt" href="hermitan.html#E8:125_1_1_3"><i><font color="Green" title="E66">A14</font></i></a>, <a class="ref" href="hermitan.html#D12" target="_self" title="HERMITAN:def.12">Def12</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E6:125_1_1"/>
<font color="Maroon" title="c5">sc</font> is <a href="hermitan.html#V5" title="HERMITAN:attr.5">diagReR+0valued</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">v</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="c1">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="hermitan.html#D7" target="_self" title="HERMITAN:def.7">HERMITAN:def 7</a><br/>

<b>consider </b><font color="Maroon" title="c7">va</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E2:125_1_1_4"/><i><font color="Green" title="E63">A15</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">va</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="c2">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>reconsider </b><font color="Maroon" title="c8">A</font> = <font color="Maroon" title="c6">v</font> as   <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="c1">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="hermitan.html#T59" target="_self" title="HERMITAN:th.59">Th59</a></i>;<br/>
<a NAME="E4:125_1_1_4"/>
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">va</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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="hermitan.html#E2:125_1_1_4"><i><font color="Green" title="E63">A15</font></i></a>, <a class="ref" href="hermitan.html#T59" target="_self" title="HERMITAN:th.59">Th59</a></i>;<br/>
<a NAME="E5:125_1_1_4"/><b>then </b>
<font color="Maroon" title="c5">sc</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">va</font>,<font color="Maroon" title="c7">va</font>
 
<b>by </b><i><a class="txt" href="hermitan.html#E2:125_1_1_4"><i><font color="Green" title="E63">A15</font></i></a>, <a class="ref" href="hermitan.html#D12" target="_self" title="HERMITAN:def.12">Def12</a></i>;<br/>
<b>hence </b><a NAME="E6:125_1_1_4"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">sc</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c6">v</font></span>)</span>
 <b>by </b><i><a class="ref" href="hermitan.html#D7" target="_self" title="HERMITAN:def.7">Def7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:125_1_1"/>
 <a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c2">f</font> is  <a href="hermitan.html#V5" title="HERMITAN:attr.5">diagReR+0valued</a> <a href="hermitan.html#NM3" title="HERMITAN:NM.3">hermitan-Form</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c1">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="hermitan.html#E3:125_1_1"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="hermitan.html#E4:125_1_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="txt" href="hermitan.html#E5:125_1_1"><i><font color="Green" title="E62">A10</font></i></a></i>;<br/>


</div>
