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<b>set </b><font color="Maroon" title="c1">K</font> =  <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<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  <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</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">f</font> being  <a href="hermitan.html#NM2" title="HERMITAN:NM.2">sesquilinear-Form</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font> holds <br/> (  <a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</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="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</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="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</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="hermitan.html#NM2" title="HERMITAN:NM.2">sesquilinear-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="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</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="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</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="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span>;<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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></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"><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> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </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"><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> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </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="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</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="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:118"/><i><font color="Green" title="E58">A1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</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> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </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="E2:118"/><i><font color="Green" title="E59">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</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"><span class="p5">(<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="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </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="E3:118"/><i><font color="Green" title="E60">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="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</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="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</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> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </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="E4:118"/><i><font color="Green" title="E61">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"><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> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="hermitan.html#T64" target="_self" title="HERMITAN:th.64">Th64</a></i>;<br/>
<a NAME="E5:118"/><i><font color="Green" title="E62">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="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</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="hermitan.html#T65" target="_self" title="HERMITAN:th.65">Th65</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_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="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> implies <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</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="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</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="E1:118_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="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</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="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</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="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</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>, <font color="Maroon" title="c15">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:118_1"/><i><font color="Green" title="E63">A6</font></i>: 
( <font color="Maroon" title="c14">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> &amp; <font color="Maroon" title="c15">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">A</font>,<font color="Maroon" title="c15">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">A</font> = <font color="Maroon" title="c14">A</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> <b>by </b><i><a class="txt" href="#E3:118_1"><i><font color="Green" title="E63">A6</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">B</font> = <font color="Maroon" title="c15">B</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"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E3:118_1"><i><font color="Green" title="E63">A6</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c18">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="E7:118_1"/><i><font color="Green" title="E64">A7</font></i>: 
<font color="Maroon" title="c16">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">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="c19">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="E9:118_1"/><i><font color="Green" title="E65">A8</font></i>: 
<font color="Maroon" title="c17">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/><a NAME="E10:118_1"/><i><font color="Green" title="E66">A9</font></i>: 
 <a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span> <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"><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> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span>
 <b>by </b><i><a class="ref" href="hermitan.html#T64" target="_self" title="HERMITAN:th.64">Th64</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</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="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</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">A</font>,<font color="Maroon" title="c17">B</font>
<b>by </b><i><a class="txt" href="#E3:118_1"><i><font color="Green" title="E63">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="c18">v</font>,<font color="Maroon" title="c19">w</font>
<b>by </b><i><a class="txt" href="#E7:118_1"><i><font color="Green" title="E64">A7</font></i></a>, <a class="txt" href="#E9:118_1"><i><font color="Green" title="E65">A8</font></i></a>, <a class="ref" href="hermitan.html#D12" target="_self" title="HERMITAN:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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="c16">A</font>,<font color="Maroon" title="c19">w</font>
<b>by </b><i><a class="txt" href="#E7:118_1"><i><font color="Green" title="E64">A7</font></i></a>, <a class="ref" href="bilinear.html#D21" title="BILINEAR:def.21">BILINEAR:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</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="c16">A</font>,<font color="Maroon" title="c17">B</font>
<b>by </b><i><a class="txt" href="#E9:118_1"><i><font color="Green" title="E65">A8</font></i></a>, <a class="txt" href="#E10:118_1"><i><font color="Green" title="E66">A9</font></i></a>, <a class="ref" href="hermitan.html#T62" target="_self" title="HERMITAN:th.62">Th62</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</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="#E3:118_1"><i><font color="Green" title="E63">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="E7:118"/>
 <a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</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="#E1:118"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="#E2:118"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="#E4:118"><i><font color="Green" title="E61">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="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</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="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_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="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> implies <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</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="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</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="E1:118_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="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</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="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</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="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</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>, <font color="Maroon" title="c15">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:118_2"/><i><font color="Green" title="E63">A10</font></i>: 
( <font color="Maroon" title="c14">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> &amp; <font color="Maroon" title="c15">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">A</font>,<font color="Maroon" title="c15">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">A</font> = <font color="Maroon" title="c14">A</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> <b>by </b><i><a class="txt" href="#E3:118_2"><i><font color="Green" title="E63">A10</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">B</font> = <font color="Maroon" title="c15">B</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"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E3:118_2"><i><font color="Green" title="E63">A10</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c18">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="E7:118_2"/><i><font color="Green" title="E64">A11</font></i>: 
<font color="Maroon" title="c16">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">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="c19">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="E9:118_2"/><i><font color="Green" title="E65">A12</font></i>: 
<font color="Maroon" title="c17">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/><a NAME="E10:118_2"/><i><font color="Green" title="E66">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="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="hermitan.html#T65" target="_self" title="HERMITAN:th.65">Th65</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</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="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</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">A</font>,<font color="Maroon" title="c17">B</font>
<b>by </b><i><a class="txt" href="#E3:118_2"><i><font color="Green" title="E63">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="c18">v</font>,<font color="Maroon" title="c19">w</font>
<b>by </b><i><a class="txt" href="#E7:118_2"><i><font color="Green" title="E64">A11</font></i></a>, <a class="txt" href="#E9:118_2"><i><font color="Green" title="E65">A12</font></i></a>, <a class="ref" href="hermitan.html#D12" target="_self" title="HERMITAN:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</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="c18">v</font>,<font color="Maroon" title="c17">B</font>
<b>by </b><i><a class="txt" href="#E9:118_2"><i><font color="Green" title="E65">A12</font></i></a>, <a class="ref" href="hermitan.html#T62" target="_self" title="HERMITAN:th.62">Th62</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="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</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="c16">A</font>,<font color="Maroon" title="c17">B</font>
<b>by </b><i><a class="txt" href="#E7:118_2"><i><font color="Green" title="E64">A11</font></i></a>, <a class="txt" href="#E10:118_2"><i><font color="Green" title="E66">A13</font></i></a>, <a class="ref" href="bilinear.html#D21" title="BILINEAR:def.21">BILINEAR:def 21</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="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</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="#E3:118_2"><i><font color="Green" title="E63">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="E9:118"/>
 <a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</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="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:118"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="#E3:118"><i><font color="Green" title="E60">A3</font></i></a>, <a class="txt" href="#E5:118"><i><font color="Green" title="E62">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/>


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