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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><a NAME="E1:55_1_1"/><i><font color="Green" title="E28">A1</font></i>: 
( <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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>let </b><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>; <i><font color="Red">:: according to </font></i><a class="ref" href="bilinear.html#D12" title="BILINEAR:def.12">BILINEAR:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="hahnban1.html#V1" title="HAHNBAN1:attr.1">additive</a></span><br/>

<b>set </b><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/>
<b>set </b><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/>
<a NAME="E2:55_1_1"/><i><font color="Green" title="E29">A2</font></i>: 
 <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> is <a href="hahnban1.html#V1" title="HAHNBAN1:attr.1">additive</a>
 
<b>by </b><i><a class="txt" href="#E1:55_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>thus </b><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>
<b>by </b><i><a class="ref" href="bilinear.html#T9" title="BILINEAR:th.9">BILINEAR:9</a></i>
<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> 
<b>by </b><i><a class="txt" href="#E1:55_1_1"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="hermitan.html#D5" target="_self" title="HERMITAN:def.5">Def5</a></i>
<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> 
<b>by </b><i><a class="ref" href="bilinear.html#T10" title="BILINEAR:th.10">BILINEAR:10</a></i>
<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> 
<b>by </b><i><a class="txt" href="#E2:55_1_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="hahnban1.html#D11" title="HAHNBAN1:def.11">HAHNBAN1:def 11</a></i>
<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> 
<b>by </b><i><a class="ref" href="bilinear.html#T10" title="BILINEAR:th.10">BILINEAR:10</a></i>
<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> 
<b>by </b><i><a class="ref" href="bilinear.html#T10" title="BILINEAR:th.10">BILINEAR:10</a></i>
<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>
<b>by </b><i><a class="ref" href="complfld.html#T87" title="COMPLFLD:th.87">COMPLFLD:87</a></i>
<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>
<b>by </b><i><a class="txt" href="#E1:55_1_1"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="hermitan.html#D5" target="_self" title="HERMITAN:def.5">Def5</a></i>
<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>
<b>by </b><i><a class="txt" href="#E1:55_1_1"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="hermitan.html#D5" target="_self" title="HERMITAN:def.5">Def5</a></i>
<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>
<b>by </b><i><a class="ref" href="bilinear.html#T9" title="BILINEAR:th.9">BILINEAR:9</a></i>
<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>
<b>by </b><i><a class="ref" href="bilinear.html#T9" title="BILINEAR:th.9">BILINEAR:9</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:55_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="hahnban1.html#V1" title="HAHNBAN1:attr.1">additive</a>
 <b>by </b><i><a class="ref" href="hahnban1.html#D11" title="HAHNBAN1:def.11">HAHNBAN1:def 11</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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