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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K19" title="BILINEAR:func.19">QForm</a> <font color="Olive" title="b3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="bilinear.html#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="Olive" title="b3">f</font></span>)</span> &amp;  <a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Olive" title="b3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="bilinear.html#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="Olive" title="b3">f</font></span>)</span> )</span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K19" title="BILINEAR:func.19">QForm</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="bilinear.html#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="Olive" title="b1">f</font></span>)</span> &amp;  <a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="bilinear.html#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="Olive" title="b1">f</font></span>)</span> )</span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="bilinear.html#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> &amp;  <a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="bilinear.html#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> )</span><br/>





<b>set </b><font color="Maroon" title="c5">L</font> =  <a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font>;<br/>
<b>set </b><font color="Maroon" title="c6">vq</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">R</font> =  <a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font>;<br/>
<b>set </b><font color="Maroon" title="c8">wq</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">RL</font> =  <a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</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/>
<b>set </b><font color="Maroon" title="c10">wqr</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p1">(<span class="default"><a href="bilinear.html#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/>
<b>set </b><font color="Maroon" title="c11">LR</font> =  <a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</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/>
<b>set </b><font color="Maroon" title="c12">vql</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#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/>
<a NAME="E1:114"/><i><font color="Green" title="E51">A1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p3">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p3">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:114"/><i><font color="Green" title="E52">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p3">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p3">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p4">(<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><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:114"/><i><font color="Green" title="E53">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p3">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <span class="p4">(<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>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p3">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:114"/><i><font color="Green" title="E54">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="bilinear.html#T47" target="_self" title="BILINEAR:th.47">Th47</a></i>;<br/>
<a NAME="E5:114"/><i><font color="Green" title="E55">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="bilinear.html#T48" target="_self" title="BILINEAR:th.48">Th48</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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> implies <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bilinear.html#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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> )</span><br/><b>assume </b><a NAME="E1:114_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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:()"><i><font color="Red">::  thesis: </font></i></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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bilinear.html#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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span><br/><b>then consider </b><font color="Maroon" title="c14">A</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>, <font color="Maroon" title="c15">B</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:114_1"/><i><font color="Green" title="E56">A6</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">A</font>,<font color="Maroon" title="c15">B</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">v</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E5:114_1"/><i><font color="Green" title="E57">A7</font></i>: 
<font color="Maroon" title="c14">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">w</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font><b> such that </b><br/><a NAME="E7:114_1"/><i><font color="Green" title="E58">A8</font></i>: 
<font color="Maroon" title="c15">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">w</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/><a NAME="E8:114_1"/><i><font color="Green" title="E59">A9</font></i>: 
 <a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</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>
 <b>by </b><i><a class="ref" href="bilinear.html#T47" target="_self" title="BILINEAR:th.47">Th47</a></i>;<br/><b>thus </b><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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#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">A</font>,<font color="Maroon" title="c15">B</font>
<b>by </b><i><a class="txt" href="#E3:114_1"><i><font color="Green" title="E56">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">v</font>,<font color="Maroon" title="c17">w</font>
<b>by </b><i><a class="txt" href="#E5:114_1"><i><font color="Green" title="E57">A7</font></i></a>, <a class="txt" href="#E7:114_1"><i><font color="Green" title="E58">A8</font></i></a>, <a class="ref" href="bilinear.html#D23" target="_self" title="BILINEAR:def.23">Def23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c14">A</font>,<font color="Maroon" title="c17">w</font>
<b>by </b><i><a class="txt" href="#E5:114_1"><i><font color="Green" title="E57">A7</font></i></a>, <a class="ref" href="bilinear.html#D21" target="_self" title="BILINEAR:def.21">Def21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">A</font>,<font color="Maroon" title="c15">B</font>
<b>by </b><i><a class="txt" href="#E7:114_1"><i><font color="Green" title="E58">A8</font></i></a>, <a class="txt" href="#E8:114_1"><i><font color="Green" title="E59">A9</font></i></a>, <a class="ref" href="bilinear.html#D22" target="_self" title="BILINEAR:def.22">Def22</a></i>
<br/>.= 
<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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
<b>by </b><i><a class="txt" href="#E3:114_1"><i><font color="Green" title="E56">A6</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:114"/>
 <a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="bilinear.html#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>
 <b>by </b><i><a class="txt" href="#E1:114"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E2:114"><i><font color="Green" title="E52">A2</font></i></a>, <a class="txt" href="#E4:114"><i><font color="Green" title="E54">A4</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="bilinear.html#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></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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> implies <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bilinear.html#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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> )</span><br/><b>assume </b><a NAME="E1:114_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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:()"><i><font color="Red">::  thesis: </font></i></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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bilinear.html#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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span><br/><b>then consider </b><font color="Maroon" title="c14">A</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>, <font color="Maroon" title="c15">B</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:114_2"/><i><font color="Green" title="E56">A10</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">A</font>,<font color="Maroon" title="c15">B</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">v</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E5:114_2"/><i><font color="Green" title="E57">A11</font></i>: 
<font color="Maroon" title="c14">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">w</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font><b> such that </b><br/><a NAME="E7:114_2"/><i><font color="Green" title="E58">A12</font></i>: 
<font color="Maroon" title="c15">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">w</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/><a NAME="E8:114_2"/><i><font color="Green" title="E59">A13</font></i>: 
 <a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</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>
 <b>by </b><i><a class="ref" href="bilinear.html#T48" target="_self" title="BILINEAR:th.48">Th48</a></i>;<br/><b>thus </b><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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#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">A</font>,<font color="Maroon" title="c15">B</font>
<b>by </b><i><a class="txt" href="#E3:114_2"><i><font color="Green" title="E56">A10</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">v</font>,<font color="Maroon" title="c17">w</font>
<b>by </b><i><a class="txt" href="#E5:114_2"><i><font color="Green" title="E57">A11</font></i></a>, <a class="txt" href="#E7:114_2"><i><font color="Green" title="E58">A12</font></i></a>, <a class="ref" href="bilinear.html#D23" target="_self" title="BILINEAR:def.23">Def23</a></i>
<br/>.= 
<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> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">v</font>,<font color="Maroon" title="c15">B</font>
<b>by </b><i><a class="txt" href="#E7:114_2"><i><font color="Green" title="E58">A12</font></i></a>, <a class="ref" href="bilinear.html#D22" target="_self" title="BILINEAR:def.22">Def22</a></i>
<br/>.= 
<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#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">A</font>,<font color="Maroon" title="c15">B</font>
<b>by </b><i><a class="txt" href="#E5:114_2"><i><font color="Green" title="E57">A11</font></i></a>, <a class="txt" href="#E8:114_2"><i><font color="Green" title="E59">A13</font></i></a>, <a class="ref" href="bilinear.html#D21" target="_self" title="BILINEAR:def.21">Def21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
<b>by </b><i><a class="txt" href="#E3:114_2"><i><font color="Green" title="E56">A10</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:114"/>
 <a href="bilinear.html#K19" title="BILINEAR:func.19">QForm</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="bilinear.html#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>
 <b>by </b><i><a class="txt" href="#E1:114"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E3:114"><i><font color="Green" title="E53">A3</font></i></a>, <a class="txt" href="#E5:114"><i><font color="Green" title="E55">A5</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
