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<span class="kw">set </span><font color="Maroon" title="c1">K</font> =  <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c2">V</font>, <font color="Maroon" title="c3">W</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of  <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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#K7" title="HERMITAN:func.7">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#K6" title="HERMITAN:func.6">RQ*Form</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Olive" title="b1">f</font></span>)</span> &amp;  <a href="hermitan.html#K7" title="HERMITAN:func.7">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#K15" title="BILINEAR:func.15">LQForm</a> <span class="p1">(<span class="default"><a href="hermitan.html#K6" title="HERMITAN:func.6">RQ*Form</a> <font color="Olive" title="b1">f</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="hermitan.html#K7" title="HERMITAN:func.7">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#K6" title="HERMITAN:func.6">RQ*Form</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="hermitan.html#K7" title="HERMITAN:func.7">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#K15" title="BILINEAR:func.15">LQForm</a> <span class="p1">(<span class="default"><a href="hermitan.html#K6" title="HERMITAN:func.6">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c5">L</font> =  <a href="bilinear.html#K13" title="BILINEAR:func.13">LKer</a> <font color="Maroon" title="c4">f</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">vq</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K13" title="BILINEAR:func.13">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c7">R</font> =  <a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <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/>
<span class="kw">set </span><font color="Maroon" title="c8">wq</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">W</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <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/>
<span class="kw">set </span><font color="Maroon" title="c9">RL</font> =  <a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">wqr</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">W</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a></span>)</span></span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c11">LR</font> =  <a href="bilinear.html#K13" title="BILINEAR:func.13">LKer</a> <span class="p1">(<span class="default"><a href="hermitan.html#K6" title="HERMITAN:func.6">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">vql</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K13" title="BILINEAR:func.13">LKer</a> <span class="p2">(<span class="default"><a href="hermitan.html#K6" title="HERMITAN:func.6">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>);<br/>
<a NAME="E1:118"/><span class="lab"><font color="Green" title="E58">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">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#K13" title="BILINEAR:func.13">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>)</span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">W</font>,<span class="p3">(<span class="default"><a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <span class="p4">(<span class="default"><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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E59">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:118_1"/><span class="lab"><font color="Green" title="E60">A3</font></span>: 
 <a href="bilinear.html#K14" title="BILINEAR:func.14">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#K14" title="BILINEAR:func.14">RKer</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#T61" target="_self" title="HERMITAN:th.61">Th64</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> implies <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">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#K6" title="HERMITAN:func.6">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> )</span><br/><span class="kw">assume </span><a NAME="E2:118_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">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#K6" title="HERMITAN:func.6">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">A</font>, <font color="Maroon" title="c15">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:118_1"/><span class="lab"><font color="Green" title="E61">A4</font></span>: 
<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#K13" title="BILINEAR:func.13">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:118_1"/><span class="lab"><font color="Green" title="E62">A5</font></span>: 
<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#K14" title="BILINEAR:func.14">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>
 <span class="kw">and </span><br/><a NAME="E6:118_1"/><span class="lab"><font color="Green" title="E63">A6</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/><span class="kw">reconsider </span><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#K13" title="BILINEAR:func.13">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:118_1"><span class="lab"><font color="Green" title="E61">A4</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">v</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E9:118_1"/><span class="lab"><font color="Green" title="E64">A7</font></span>: 
<font color="Maroon" title="c16">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K13" title="BILINEAR:func.13">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#T22" title="VECTSP10:th.22">VECTSP10:22</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">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#K14" title="BILINEAR:func.14">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:118_1"><span class="lab"><font color="Green" title="E62">A5</font></span></a></span>;<br/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E12:118_1"/><span class="lab"><font color="Green" title="E65">A8</font></span>: 
<font color="Maroon" title="c18">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#K14" title="BILINEAR:func.14">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#T22" title="VECTSP10:th.22">VECTSP10:22</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">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#K7" title="HERMITAN:func.7">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="c18">B</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:118_1"><span class="lab"><font color="Green" title="E63">A6</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c17">v</font>,<font color="Maroon" title="c19">w</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:118_1"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E12:118_1"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="ref" href="hermitan.html#D12" target="_self" title="HERMITAN:def.12">Def12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c16">A</font>,<font color="Maroon" title="c19">w</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:118_1"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="ref" href="bilinear.html#D20" title="BILINEAR:def.20">BILINEAR:def 20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="hermitan.html#K6" title="HERMITAN:func.6">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c16">A</font>,<font color="Maroon" title="c18">B</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:118_1"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="txt" href="#E1:118_1"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="ref" href="hermitan.html#T59" target="_self" title="HERMITAN:th.59">Th62</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="hermitan.html#K6" title="HERMITAN:func.6">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:118_1"><span class="lab"><font color="Green" title="E63">A6</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:118"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="hermitan.html#K6" title="HERMITAN:func.6">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<span class="p3">(<span class="default"><a href="bilinear.html#K13" title="BILINEAR:func.13">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>)</span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">W</font>,<span class="p3">(<span class="default"><a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="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> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <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#K14" title="BILINEAR:func.14">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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#T61" target="_self" title="HERMITAN:th.61">Th64</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:118"/>
 <a href="hermitan.html#K7" title="HERMITAN:func.7">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#K6" title="HERMITAN:func.6">RQ*Form</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E2:118"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="hermitan.html#K7" title="HERMITAN:func.7">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#K15" title="BILINEAR:func.15">LQForm</a> <span class="p1">(<span class="default"><a href="hermitan.html#K6" title="HERMITAN:func.6">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span><br/>

<div><span class="lab"><font color="Green" title="E60">A9</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="118_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:118_2"/><span class="lab"><font color="Green" title="E61">A10</font></span>: 
 <a href="bilinear.html#K13" title="BILINEAR:func.13">LKer</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">LKer</a> <span class="p1">(<span class="default"><a href="hermitan.html#K6" title="HERMITAN:func.6">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#T62" target="_self" title="HERMITAN:th.62">Th65</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> implies <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">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#K15" title="BILINEAR:func.15">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K6" title="HERMITAN:func.6">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/><span class="kw">assume </span><a NAME="E2:118_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">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#K15" title="BILINEAR:func.15">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K6" title="HERMITAN:func.6">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/><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E4:118_2"/><span class="lab"><font color="Green" title="E62">A11</font></span>: 
<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#K13" title="BILINEAR:func.13">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:118_2"/><span class="lab"><font color="Green" title="E63">A12</font></span>: 
<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#K14" title="BILINEAR:func.14">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>
 <span class="kw">and </span><br/><a NAME="E6:118_2"/><span class="lab"><font color="Green" title="E64">A13</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/><span class="kw">reconsider </span><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#K13" title="BILINEAR:func.13">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:118_2"><span class="lab"><font color="Green" title="E62">A11</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">v</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E9:118_2"/><span class="lab"><font color="Green" title="E65">A14</font></span>: 
<font color="Maroon" title="c16">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K13" title="BILINEAR:func.13">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#T22" title="VECTSP10:th.22">VECTSP10:22</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">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#K14" title="BILINEAR:func.14">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:118_2"><span class="lab"><font color="Green" title="E63">A12</font></span></a></span>;<br/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E12:118_2"/><span class="lab"><font color="Green" title="E66">A15</font></span>: 
<font color="Maroon" title="c18">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#K14" title="BILINEAR:func.14">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#T22" title="VECTSP10:th.22">VECTSP10:22</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">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#K7" title="HERMITAN:func.7">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="c18">B</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:118_2"><span class="lab"><font color="Green" title="E64">A13</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c17">v</font>,<font color="Maroon" title="c19">w</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:118_2"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="txt" href="#E12:118_2"><span class="lab"><font color="Green" title="E66">A15</font></span></a>, <a class="ref" href="hermitan.html#D12" target="_self" title="HERMITAN:def.12">Def12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="hermitan.html#K6" title="HERMITAN:func.6">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="c17">v</font>,<font color="Maroon" title="c18">B</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:118_2"><span class="lab"><font color="Green" title="E66">A15</font></span></a>, <a class="ref" href="hermitan.html#T59" target="_self" title="HERMITAN:th.59">Th62</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K6" title="HERMITAN:func.6">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="c18">B</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:118_2"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="txt" href="#E1:118_2"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="ref" href="bilinear.html#D20" title="BILINEAR:def.20">BILINEAR:def 20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K6" title="HERMITAN:func.6">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:118_2"><span class="lab"><font color="Green" title="E64">A13</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:118"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K6" title="HERMITAN:func.6">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#K13" title="BILINEAR:func.13">LKer</a> <span class="p4">(<span class="default"><a href="hermitan.html#K6" title="HERMITAN:func.6">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#K14" title="BILINEAR:func.14">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> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K13" title="BILINEAR:func.13">LKer</a> <span class="p3">(<span class="default"><a href="hermitan.html#K6" title="HERMITAN:func.6">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#K13" title="BILINEAR:func.13">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#T62" target="_self" title="HERMITAN:th.62">Th65</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:118"/>
 <a href="hermitan.html#K7" title="HERMITAN:func.7">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#K15" title="BILINEAR:func.15">LQForm</a> <span class="p1">(<span class="default"><a href="hermitan.html#K6" title="HERMITAN:func.6">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E5:118"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
