<?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="Red">::  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#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Olive" title="b3">f</font></span>)</span></span>)</span> &amp;  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Olive" title="b3">f</font></span>)</span></span>)</span> &amp;  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Olive" title="b3">f</font></span>)</span></span>)</span> &amp;  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Olive" title="b3">f</font></span>)</span></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="Red">::  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#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> &amp;  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> &amp;  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> &amp;  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Olive" title="b1">f</font></span>)</span></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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span> &amp;  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span> &amp;  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> &amp;  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c5">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="c6">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="c7">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#K16" title="BILINEAR:func.16">RKer</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></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">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#K15" title="BILINEAR:func.15">LKer</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">rlf</font> =  <a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">qf</font> =  <a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">lrf</font> =  <a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>;<br/>
<span class="kw">thus </span><a NAME="E1:116"/>
 <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <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#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span> &amp;  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> &amp;  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:116_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</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="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">vv</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><span class="kw"> such that </span><br/><a NAME="E3:116_1"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">vv</font>
 <span class="kw">and </span><br/><a NAME="E4:116_1"/><span class="lab"><font color="Green" title="E53">A2</font></span>: 
for <font color="Olive" title="b1">ww</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  <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">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="c13">vv</font>,<font color="Olive" title="b1">ww</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">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> <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></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#K18" title="BILINEAR:func.18">RQForm</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></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">vv</font>,<font color="Maroon" title="c14">ww</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></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c15">w</font> = <font color="Maroon" title="c14">ww</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="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="bilinear.html#T47" target="_self" title="BILINEAR:th.47">Th47</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">vv</font>,<font color="Maroon" title="c14">ww</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">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="c13">vv</font>,<font color="Maroon" title="c15">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T49" target="_self" title="BILINEAR:th.49">Th49</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_1"><span class="lab"><font color="Green" title="E53">A2</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:116_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:116_1"><span class="lab"><font color="Green" title="E52">A1</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/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:116"/>
 <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</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"> (  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span> &amp;  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> &amp;  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:116_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></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="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">vv</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><span class="kw"> such that </span><br/><a NAME="E3:116_2"/><span class="lab"><font color="Green" title="E52">A3</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">vv</font>
 <span class="kw">and </span><br/><a NAME="E4:116_2"/><span class="lab"><font color="Green" title="E53">A4</font></span>: 
for <font color="Olive" title="b1">ww</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> <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></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">vv</font>,<font color="Olive" title="b1">ww</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">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"> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">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="c13">vv</font>,<font color="Maroon" title="c14">ww</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></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c15">w</font> = <font color="Maroon" title="c14">ww</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="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</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></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">Th47</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">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="c13">vv</font>,<font color="Maroon" title="c14">ww</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">vv</font>,<font color="Maroon" title="c15">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T49" target="_self" title="BILINEAR:th.49">Th49</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_2"><span class="lab"><font color="Green" title="E53">A4</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:116_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:116_2"><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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E3:116"/>
 <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <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#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> &amp;  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:116_3"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</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="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">ww</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><span class="kw"> such that </span><br/><a NAME="E3:116_3"/><span class="lab"><font color="Green" title="E52">A5</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ww</font>
 <span class="kw">and </span><br/><a NAME="E4:116_3"/><span class="lab"><font color="Green" title="E53">A6</font></span>: 
for <font color="Olive" title="b1">vv</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> holds  <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">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="Olive" title="b1">vv</font>,<font color="Maroon" title="c13">ww</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">reconsider </span><font color="Maroon" title="c14">w</font> = <font color="Maroon" title="c13">ww</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="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</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></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">Th47</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">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"> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c15">vv</font>,<font color="Maroon" title="c14">w</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></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c15">vv</font>,<font color="Maroon" title="c14">w</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">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="c15">vv</font>,<font color="Maroon" title="c13">ww</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T49" target="_self" title="BILINEAR:th.49">Th49</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_3"><span class="lab"><font color="Green" title="E53">A6</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:116_3"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:116_3"><span class="lab"><font color="Green" title="E52">A5</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/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E4:116"/>
 <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</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"> (  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> &amp;  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:116_4"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></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="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">ww</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> <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></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:116_4"/><span class="lab"><font color="Green" title="E52">A7</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ww</font>
 <span class="kw">and </span><br/><a NAME="E4:116_4"/><span class="lab"><font color="Green" title="E53">A8</font></span>: 
for <font color="Olive" title="b1">vv</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> holds  <span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">vv</font>,<font color="Maroon" title="c13">ww</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">reconsider </span><font color="Maroon" title="c14">w</font> = <font color="Maroon" title="c13">ww</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="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="bilinear.html#T47" target="_self" title="BILINEAR:th.47">Th47</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">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"> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">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="c15">vv</font>,<font color="Maroon" title="c14">w</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></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">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="c15">vv</font>,<font color="Maroon" title="c14">w</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c15">vv</font>,<font color="Maroon" title="c13">ww</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T49" target="_self" title="BILINEAR:th.49">Th49</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_4"><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="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="E7:116_4"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:116_4"><span class="lab"><font color="Green" title="E52">A7</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/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E5:116"/>
 <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <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#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:116_5"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</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="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">vv</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><span class="kw"> such that </span><br/><a NAME="E3:116_5"/><span class="lab"><font color="Green" title="E52">A9</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">vv</font>
 <span class="kw">and </span><br/><a NAME="E4:116_5"/><span class="lab"><font color="Green" title="E53">A10</font></span>: 
for <font color="Olive" title="b1">ww</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  <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">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="c13">vv</font>,<font color="Olive" title="b1">ww</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">reconsider </span><font color="Maroon" title="c14">v</font> = <font color="Maroon" title="c13">vv</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="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <span class="p3">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T48" target="_self" title="BILINEAR:th.48">Th48</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">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"> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c14">v</font>,<font color="Maroon" title="c15">ww</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></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c14">v</font>,<font color="Maroon" title="c15">ww</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">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="c13">vv</font>,<font color="Maroon" title="c15">ww</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T49" target="_self" title="BILINEAR:th.49">Th49</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_5"><span class="lab"><font color="Green" title="E53">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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:116_5"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:116_5"><span class="lab"><font color="Green" title="E52">A9</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/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E6:116"/>
 <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</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">  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:116_6"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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"> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">vv</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> <span class="p3">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:116_6"/><span class="lab"><font color="Green" title="E52">A11</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">vv</font>
 <span class="kw">and </span><br/><a NAME="E4:116_6"/><span class="lab"><font color="Green" title="E53">A12</font></span>: 
for <font color="Olive" title="b1">ww</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  <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">vv</font>,<font color="Olive" title="b1">ww</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">reconsider </span><font color="Maroon" title="c14">v</font> = <font color="Maroon" title="c13">vv</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="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="bilinear.html#T48" target="_self" title="BILINEAR:th.48">Th48</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_6_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">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"> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">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">v</font>,<font color="Maroon" title="c15">ww</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></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">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">v</font>,<font color="Maroon" title="c15">ww</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">vv</font>,<font color="Maroon" title="c15">ww</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T49" target="_self" title="BILINEAR:th.49">Th49</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_6"><span class="lab"><font color="Green" title="E53">A12</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:116_6"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:116_6"><span class="lab"><font color="Green" title="E52">A11</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/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E7:116"/>
 <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <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#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:116_7"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</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="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">ww</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><span class="kw"> such that </span><br/><a NAME="E3:116_7"/><span class="lab"><font color="Green" title="E52">A13</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ww</font>
 <span class="kw">and </span><br/><a NAME="E4:116_7"/><span class="lab"><font color="Green" title="E53">A14</font></span>: 
for <font color="Olive" title="b1">vv</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> holds  <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">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="Olive" title="b1">vv</font>,<font color="Maroon" title="c13">ww</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_7_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">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> <span class="p3">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></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#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c14">vv</font>,<font color="Maroon" title="c13">ww</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></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c15">v</font> = <font color="Maroon" title="c14">vv</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="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="bilinear.html#T48" target="_self" title="BILINEAR:th.48">Th48</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c14">vv</font>,<font color="Maroon" title="c13">ww</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">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="c15">v</font>,<font color="Maroon" title="c13">ww</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T49" target="_self" title="BILINEAR:th.49">Th49</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_7"><span class="lab"><font color="Green" title="E53">A14</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:116_7"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:116_7"><span class="lab"><font color="Green" title="E52">A13</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/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E8:116"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</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"> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">ww</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><span class="kw"> such that </span><br/><a NAME="E10:116"/><span class="lab"><font color="Green" title="E52">A15</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ww</font>
 <span class="kw">and </span><br/><a NAME="E11:116"/><span class="lab"><font color="Green" title="E53">A16</font></span>: 
for <font color="Olive" title="b1">vv</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> <span class="p3">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">vv</font>,<font color="Maroon" title="c13">ww</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_8"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">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"> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">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">vv</font>,<font color="Maroon" title="c13">ww</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></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c15">v</font> = <font color="Maroon" title="c14">vv</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="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <span class="p3">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T48" target="_self" title="BILINEAR:th.48">Th48</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">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">vv</font>,<font color="Maroon" title="c13">ww</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" title="BILINEAR:func.18">RQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c15">v</font>,<font color="Maroon" title="c13">ww</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T49" target="_self" title="BILINEAR:th.49">Th49</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:116"><span class="lab"><font color="Green" title="E53">A16</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E13:116"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:116"><span class="lab"><font color="Green" title="E52">A15</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/>


</div>
