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<span class="kw">let </span><font color="Maroon" title="c6">w</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="bilinear.html#D11" target="_self" onmouseover="rs('bilinear/D11')" onmouseout="rh()">BILINEAR:def 11</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="bilinear.html#K7" title="BILINEAR:func.7">FunctionalFAF</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bilinear.html#K3" title="BILINEAR:func.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">w</font>) is  <a href="vectsp_1.html#V13" title="VECTSP_1:attr.13">additive</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">Ffg</font> =  <a href="bilinear.html#K7" title="BILINEAR:func.7">FunctionalFAF</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bilinear.html#K3" title="BILINEAR:func.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">w</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c8">Ff</font> =  <a href="bilinear.html#K7" title="BILINEAR:func.7">FunctionalFAF</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">w</font>);<br/>
<span class="kw">let </span><font color="Maroon" title="c9">v</font>, <font color="Maroon" title="c10">y</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c3">W</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_1.html#D19" onmouseover="rs('vectsp_1/D19')" onmouseout="rh()">VECTSP_1:def 19</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="bilinear.html#K7" title="BILINEAR:func.7">FunctionalFAF</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bilinear.html#K3" title="BILINEAR:func.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">w</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="c9">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">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#K7" title="BILINEAR:func.7">FunctionalFAF</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bilinear.html#K3" title="BILINEAR:func.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">w</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K7" title="BILINEAR:func.7">FunctionalFAF</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bilinear.html#K3" title="BILINEAR:func.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">w</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font></span>)</span></span><br/>

<a NAME="E1:60_1_1"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
 <a href="bilinear.html#K7" title="BILINEAR:func.7">FunctionalFAF</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">w</font>) is  <a href="vectsp_1.html#V13" title="VECTSP_1:attr.13">additive</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#D11" target="_self" onmouseover="rs('bilinear/D11')" onmouseout="rh()">Def11</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="bilinear.html#K7" title="BILINEAR:func.7">FunctionalFAF</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bilinear.html#K3" title="BILINEAR:func.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">w</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="c9">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <span class="p2">(<span class="default"><a href="bilinear.html#K7" title="BILINEAR:func.7">FunctionalFAF</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">w</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T15" target="_self" onmouseover="rs('bilinear/T15')" onmouseout="rh()">Th15</a></span>
<br/>.= 
<font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K7" title="BILINEAR:func.7">FunctionalFAF</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">w</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="c9">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D6" onmouseover="rs('hahnban1/D6')" onmouseout="rh()">HAHNBAN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K7" title="BILINEAR:func.7">FunctionalFAF</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">w</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K7" title="BILINEAR:func.7">FunctionalFAF</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">w</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K7" title="BILINEAR:func.7">FunctionalFAF</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">w</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">v</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K7" title="BILINEAR:func.7">FunctionalFAF</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">w</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D2" onmouseover="rs('vectsp_1/D2')" onmouseout="rh()">VECTSP_1:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <span class="p3">(<span class="default"><a href="bilinear.html#K7" title="BILINEAR:func.7">FunctionalFAF</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">w</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K7" title="BILINEAR:func.7">FunctionalFAF</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">w</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D6" onmouseover="rs('hahnban1/D6')" onmouseout="rh()">HAHNBAN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <span class="p3">(<span class="default"><a href="bilinear.html#K7" title="BILINEAR:func.7">FunctionalFAF</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">w</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <span class="p3">(<span class="default"><a href="bilinear.html#K7" title="BILINEAR:func.7">FunctionalFAF</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">w</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D6" onmouseover="rs('hahnban1/D6')" onmouseout="rh()">HAHNBAN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K7" title="BILINEAR:func.7">FunctionalFAF</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bilinear.html#K3" title="BILINEAR:func.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">w</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <span class="p3">(<span class="default"><a href="bilinear.html#K7" title="BILINEAR:func.7">FunctionalFAF</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">w</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T15" target="_self" onmouseover="rs('bilinear/T15')" onmouseout="rh()">Th15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K7" title="BILINEAR:func.7">FunctionalFAF</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bilinear.html#K3" title="BILINEAR:func.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">w</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K7" title="BILINEAR:func.7">FunctionalFAF</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bilinear.html#K3" title="BILINEAR:func.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">w</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T15" target="_self" onmouseover="rs('bilinear/T15')" onmouseout="rh()">Th15</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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