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

<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font>, <font color="Olive" title="b2">W</font> being   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font><br/>  for <font color="Olive" title="b3">f</font> being   <a href="bilinear.html#NM2" title="BILINEAR:NM.2">bilinear-Form</a> of <font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">W</font> holds <br/> (  <a href="bilinear.html#K17" title="BILINEAR:func.17">QForm</a> <font color="Olive" title="b3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Olive" title="b3">f</font></span>)</span> &amp;  <a href="bilinear.html#K17" title="BILINEAR:func.17">QForm</a> <font color="Olive" title="b3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</a> <font color="Olive" title="b3">f</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font>, <font color="Maroon" title="c3">W</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="bilinear.html#NM2" title="BILINEAR:NM.2">bilinear-Form</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font> holds <br/> (  <a href="bilinear.html#K17" title="BILINEAR:func.17">QForm</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Olive" title="b1">f</font></span>)</span> &amp;  <a href="bilinear.html#K17" title="BILINEAR:func.17">QForm</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</a> <font color="Olive" title="b1">f</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="bilinear.html#NM2" title="BILINEAR:NM.2">bilinear-Form</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="bilinear.html#K17" title="BILINEAR:func.17">QForm</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="bilinear.html#K17" title="BILINEAR:func.17">QForm</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c5">L</font> =  <a href="bilinear.html#K13" title="BILINEAR:func.13">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#K13" title="BILINEAR:func.13">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c7">R</font> =  <a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <font color="Maroon" title="c4">f</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">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#K14" title="BILINEAR:func.14">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c9">RL</font> =  <a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">wqr</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#K14" title="BILINEAR:func.14">RKer</a> <span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c11">LR</font> =  <a href="bilinear.html#K13" title="BILINEAR:func.13">LKer</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">vql</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#K13" title="BILINEAR:func.13">LKer</a> <span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>);<br/>
<a NAME="E1:113"/><span class="lab"><font color="Green" title="E50">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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#K13" title="BILINEAR:func.13">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#K14" title="BILINEAR:func.14">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E51">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> implies <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:113_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">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#K13" title="BILINEAR:func.13">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>)</span>)</span>, <font color="Maroon" title="c15">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#K14" title="BILINEAR:func.14">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E3:113_1"/><span class="lab"><font color="Green" title="E52">A3</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">A</font>,<font color="Maroon" title="c15">B</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="domain_1.html#T1" title="DOMAIN_1:th.1">DOMAIN_1:1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c16">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="E5:113_1"/><span class="lab"><font color="Green" title="E53">A4</font></span>: 
<font color="Maroon" title="c15">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">w</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K14" title="BILINEAR:func.14">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#T22" title="VECTSP10:th.22">VECTSP10:22</a></span>;<br/><a NAME="E6:113_1"/><span class="lab"><font color="Green" title="E54">A5</font></span>: 
 <a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T46" target="_self" title="BILINEAR:th.46">Th47</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">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="E8:113_1"/><span class="lab"><font color="Green" title="E55">A6</font></span>: 
<font color="Maroon" title="c14">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K13" title="BILINEAR:func.13">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#T22" title="VECTSP10:th.22">VECTSP10:22</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">QForm</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="c14">A</font>,<font color="Maroon" title="c15">B</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:113_1"><span class="lab"><font color="Green" title="E52">A3</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="c17">v</font>,<font color="Maroon" title="c16">w</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:113_1"><span class="lab"><font color="Green" title="E55">A6</font></span></a>, <a class="txt" href="#E5:113_1"><span class="lab"><font color="Green" title="E53">A4</font></span></a>, <a class="ref" href="bilinear.html#D22" target="_self" title="BILINEAR:def.22">Def23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">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="c14">A</font>,<font color="Maroon" title="c16">w</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:113_1"><span class="lab"><font color="Green" title="E55">A6</font></span></a>, <a class="ref" href="bilinear.html#D20" target="_self" title="BILINEAR:def.20">Def21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</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="c14">A</font>,<font color="Maroon" title="c15">B</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:113_1"><span class="lab"><font color="Green" title="E53">A4</font></span></a>, <a class="txt" href="#E6:113_1"><span class="lab"><font color="Green" title="E54">A5</font></span></a>, <a class="ref" href="bilinear.html#D21" target="_self" title="BILINEAR:def.21">Def22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:113_1"><span class="lab"><font color="Green" title="E52">A3</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:113"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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#K13" title="BILINEAR:func.13">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#K14" title="BILINEAR:func.14">RKer</a> <span class="p4">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  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#K14" title="BILINEAR:func.14">RKer</a> <span class="p3">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>)</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#K14" title="BILINEAR:func.14">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="bilinear.html#T46" target="_self" title="BILINEAR:th.46">Th47</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:113"/>
 <a href="bilinear.html#K17" title="BILINEAR:func.17">QForm</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:113"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="txt" href="#E2:113"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="bilinear.html#K17" title="BILINEAR:func.17">QForm</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span><br/>

<div><span class="lab"><font color="Green" title="E52">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="113_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> implies <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:113_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">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#K13" title="BILINEAR:func.13">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>)</span>)</span>, <font color="Maroon" title="c15">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#K14" title="BILINEAR:func.14">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E3:113_2"/><span class="lab"><font color="Green" title="E53">A8</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">A</font>,<font color="Maroon" title="c15">B</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="domain_1.html#T1" title="DOMAIN_1:th.1">DOMAIN_1:1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c16">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="E5:113_2"/><span class="lab"><font color="Green" title="E54">A9</font></span>: 
<font color="Maroon" title="c15">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">w</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K14" title="BILINEAR:func.14">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#T22" title="VECTSP10:th.22">VECTSP10:22</a></span>;<br/><a NAME="E6:113_2"/><span class="lab"><font color="Green" title="E55">A10</font></span>: 
 <a href="bilinear.html#K13" title="BILINEAR:func.13">LKer</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">LKer</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T47" target="_self" title="BILINEAR:th.47">Th48</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">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="E8:113_2"/><span class="lab"><font color="Green" title="E56">A11</font></span>: 
<font color="Maroon" title="c14">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K13" title="BILINEAR:func.13">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#T22" title="VECTSP10:th.22">VECTSP10:22</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">QForm</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="c14">A</font>,<font color="Maroon" title="c15">B</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:113_2"><span class="lab"><font color="Green" title="E53">A8</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="c17">v</font>,<font color="Maroon" title="c16">w</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:113_2"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="txt" href="#E5:113_2"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="ref" href="bilinear.html#D22" target="_self" title="BILINEAR:def.22">Def23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</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="c17">v</font>,<font color="Maroon" title="c15">B</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:113_2"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="ref" href="bilinear.html#D21" target="_self" title="BILINEAR:def.21">Def22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</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="c14">A</font>,<font color="Maroon" title="c15">B</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:113_2"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="txt" href="#E6:113_2"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="bilinear.html#D20" target="_self" title="BILINEAR:def.20">Def21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:113_2"><span class="lab"><font color="Green" title="E53">A8</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:113"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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#K13" title="BILINEAR:func.13">LKer</a> <span class="p4">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></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#K14" title="BILINEAR:func.14">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> &amp;  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#K13" title="BILINEAR:func.13">LKer</a> <span class="p3">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>)</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#K13" title="BILINEAR:func.13">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="bilinear.html#T47" target="_self" title="BILINEAR:th.47">Th48</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:113"/>
 <a href="bilinear.html#K17" title="BILINEAR:func.17">QForm</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:113"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="txt" href="#E5:113"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
