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

<b>let </b><font color="Maroon">K</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V4">associative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">K</font>, <font color="Maroon">W</font> be   <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="bilinear.html#NM2">bilinear-Form</a> of <font color="Maroon">V</font>,<font color="Maroon">W</font>;<br/>





<b>set </b><font color="Maroon">vq</font> =  <a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">wq</font> =  <a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <font color="Maroon">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">wqr</font> =  <a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">vql</font> =  <a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">rlf</font> =  <a href="bilinear.html#K18" target="_self">RQForm</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <font color="Maroon">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">qf</font> =  <a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">lrf</font> =  <a href="bilinear.html#K17" target="_self">LQForm</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span>;<br/>
<b>thus </b><a NAME="E1:115"/>
 <a href="bilinear.html#K12" target="_self">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:115_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">vv</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:115_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vv</font> &amp; ( for <font color="Olive">ww</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Olive">ww</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> ) )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">ww</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <span class="p3">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">ww</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <font color="Maroon">f</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="bilinear.html#E18"><i><font color="Green">Lemma49</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">ww</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="bilinear.html#T10" target="_self">Th10</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:115_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E2:115"/>
 <a href="bilinear.html#K12" target="_self">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:115_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">vv</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:115_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vv</font> &amp; ( for <font color="Olive">ww</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <span class="p3">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Olive">ww</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> ) )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">ww</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <font color="Maroon">f</font></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">ww</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <span class="p3">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="bilinear.html#E18"><i><font color="Green">Lemma49</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">ww</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="bilinear.html#T10" target="_self">Th10</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:115_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E3:115"/>
 <a href="bilinear.html#K13" target="_self">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="bilinear.html#K13" target="_self">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:115_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K13" target="_self">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">ww</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <font color="Maroon">f</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:115_3"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ww</font> &amp; ( for <font color="Olive">vv</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">vv</font>,<font color="Maroon">ww</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> ) )
 ;<br/>
<b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">ww</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <span class="p3">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="bilinear.html#E18"><i><font color="Green">Lemma49</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">vv</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">ww</font>
<b>by </b><i><a class="ref" href="bilinear.html#T10" target="_self">Th10</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:115_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K13" target="_self">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E4:115"/>
 <a href="bilinear.html#K13" target="_self">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="bilinear.html#K13" target="_self">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:115_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K13" target="_self">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">ww</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <span class="p3">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:115_4"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ww</font> &amp; ( for <font color="Olive">vv</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">vv</font>,<font color="Maroon">ww</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> ) )
 ;<br/>
<b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">ww</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <font color="Maroon">f</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="bilinear.html#E18"><i><font color="Green">Lemma49</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">vv</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">ww</font>
<b>by </b><i><a class="ref" href="bilinear.html#T10" target="_self">Th10</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:115_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K13" target="_self">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E5:115"/>
 <a href="bilinear.html#K12" target="_self">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:115_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">vv</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:115_5"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vv</font> &amp; ( for <font color="Olive">ww</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Olive">ww</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> ) )
 ;<br/>
<b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">vv</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <span class="p3">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="bilinear.html#T9" target="_self">Th9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">ww</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <font color="Maroon">f</font></span>)</span></span>)</span>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">ww</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">ww</font>
<b>by </b><i><a class="ref" href="bilinear.html#T10" target="_self">Th10</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:115_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E6:115"/>
 <a href="bilinear.html#K12" target="_self">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:115_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">vv</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <span class="p3">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:115_6"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vv</font> &amp; ( for <font color="Olive">ww</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Olive">ww</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> ) )
 ;<br/>
<b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">vv</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="bilinear.html#T9" target="_self">Th9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">ww</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <font color="Maroon">f</font></span>)</span></span>)</span>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">ww</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">ww</font>
<b>by </b><i><a class="ref" href="bilinear.html#T10" target="_self">Th10</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:115_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E7:115"/>
 <a href="bilinear.html#K13" target="_self">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="bilinear.html#K13" target="_self">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:115_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K13" target="_self">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">ww</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <font color="Maroon">f</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:115_7"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ww</font> &amp; ( for <font color="Olive">vv</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">vv</font>,<font color="Maroon">ww</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> ) )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">vv</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <span class="p3">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">vv</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="bilinear.html#T9" target="_self">Th9</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">ww</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">ww</font>
<b>by </b><i><a class="ref" href="bilinear.html#T10" target="_self">Th10</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:115_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K13" target="_self">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


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

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E8:115"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K13" target="_self">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">ww</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <font color="Maroon">f</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E10:115"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ww</font> &amp; ( for <font color="Olive">vv</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <span class="p3">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">vv</font>,<font color="Maroon">ww</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> ) )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">vv</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">vv</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" target="_self">LKer</a> <span class="p3">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="bilinear.html#T9" target="_self">Th9</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">ww</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K17" target="_self">LQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">ww</font>
<b>by </b><i><a class="ref" href="bilinear.html#T10" target="_self">Th10</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:115"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K13" target="_self">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K19" target="_self">QForm</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
