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<span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="bilinear.html#NM1" title="BILINEAR:NM.1">Form</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is  <a href="hermitan.html#V3" title="HERMITAN:attr.3">hermitan</a>  &amp; <font color="Maroon" title="c2">f</font> is  <a href="bilinear.html#V2" title="BILINEAR:attr.2">additiveSAF</a>  implies <font color="Maroon" title="c2">f</font> is  <a href="bilinear.html#V1" title="BILINEAR:attr.1">additiveFAF</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:58_1_1"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
( <font color="Maroon" title="c2">f</font> is  <a href="hermitan.html#V3" title="HERMITAN:attr.3">hermitan</a>  &amp; <font color="Maroon" title="c2">f</font> is  <a href="bilinear.html#V2" title="BILINEAR:attr.2">additiveSAF</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> is  <a href="bilinear.html#V1" title="BILINEAR:attr.1">additiveFAF</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">v1</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c1">V</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="bilinear.html#D12" title="BILINEAR:def.12">BILINEAR:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v1</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">F</font> =  <a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">F2</font> =  <a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v1</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y</font></span>)</span> = 
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T9" title="BILINEAR:th.9">BILINEAR:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y</font></span>)</span>,<font color="Maroon" title="c3">v1</font></span>)</span> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="hermitan.html#D5" target="_self" title="HERMITAN:def.5">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T10" title="BILINEAR:th.10">BILINEAR:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T10" title="BILINEAR:th.10">BILINEAR:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c3">v1</font></span>)</span></span>)</span> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T10" title="BILINEAR:th.10">BILINEAR:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">v1</font></span>)</span> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c3">v1</font></span>)</span> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#T87" title="COMPLFLD:th.87">COMPLFLD:87</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c6">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c3">v1</font></span>)</span> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="hermitan.html#D5" target="_self" title="HERMITAN:def.5">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c6">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c7">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="hermitan.html#D5" target="_self" title="HERMITAN:def.5">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c7">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T9" title="BILINEAR:th.9">BILINEAR:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T9" title="BILINEAR:th.9">BILINEAR:9</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:58_1_1"/>
 <a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v1</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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