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<b>let </b><font color="Maroon">K</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a> <a href="vectsp_1.html#V9">Field-like</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">V</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">W</font> be   <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="hahnban1.html#NM1">Functional</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="hahnban1.html#NM2">linear-Functional</a> of <font color="Maroon">W</font>;<br/>









<b>assume </b><a NAME="E1:121"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="hahnban1.html#K7">0Functional</a> <font color="Maroon">V</font>
 ;<br/>

<b>set </b><font color="Maroon">fg</font> =  <a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>;<br/>
<b>set </b><font color="Maroon">cg</font> =  <a href="vectsp10.html#K11">CQFunctional</a> <font color="Maroon">g</font>;<br/>
<b>set </b><font color="Maroon">fcfg</font> =  <a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="vectsp10.html#K11">CQFunctional</a> <font color="Maroon">g</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">wqr</font> =  <a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <span class="p2">(<span class="default"><a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">wq</font> =  <a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p1">(<span class="default"><a href="vectsp10.html#K9">Ker</a> <font color="Maroon">g</font></span>)</span>;<br/>
<a NAME="E2:121"/><i><font color="Green">E37</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <span class="p2">(<span class="default"><a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="bilinear.html#K13" target="_self">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:121"/>
 <a href="bilinear.html#K13" target="_self">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp10.html#K8">ker</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="bilinear.html#T17" target="_self">Th17</a>, <a class="ref" href="bilinear.html#T14" target="_self">Th14</a></i>;<br/>
<b>hence </b><a NAME="E4:121"/><i><font color="Green">E38</font></i>: 
 <a href="bilinear.html#K16" target="_self">RKer</a> <span class="p1">(<span class="default"><a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp10.html#K9">Ker</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="bilinear.html#T18" target="_self">Th18</a>, <a class="ref" href="vectsp10.html#D11">VECTSP10:def 11</a></i>;<br/>

<a NAME="E5:121"/><i><font color="Green">E81</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p3">(<span class="default"><a href="bilinear.html#K16" target="_self">RKer</a> <span class="p4">(<span class="default"><a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:121"/><i><font color="Green">E82</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="vectsp10.html#K11">CQFunctional</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p3">(<span class="default"><a href="vectsp10.html#K9">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:121_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="vectsp10.html#K11">CQFunctional</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">A</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Maroon">B</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="vectsp10.html#K9">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:121_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/><b>consider </b><font color="Maroon">w</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E5:121_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="vectsp_4.html#K3">+</a> <span class="p1">(<span class="default"><a href="vectsp10.html#K9">Ker</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23">VECTSP10:23</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="vectsp10.html#K11">CQFunctional</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="vectsp10.html#K11">CQFunctional</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp10.html#K11">CQFunctional</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp10.html#T36">VECTSP10:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">A</font>,<font color="Maroon">w</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="bilinear.html#D11" target="_self">Def11</a>, , <a class="ref" href="bilinear.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="bilinear.html#K18" target="_self">RQForm</a> <span class="p2">(<span class="default"><a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:121"/>
 <a href="bilinear.html#K18" target="_self">RQForm</a> <span class="p1">(<span class="default"><a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="funct_2.html#R1">=</a>  <a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="vectsp10.html#K11">CQFunctional</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="bilinear.html#D11" target="_self">Def11</a>, , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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