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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="c3">W</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="bilinear.html#D14" target="_self" onmouseover="rs('bilinear/D14')" onmouseout="rh()">BILINEAR:def 14</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#K8" title="BILINEAR:func.8">FunctionalSAF</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">w</font>) is  <a href="hahnban1.html#V1" title="HAHNBAN1:attr.1">homogeneous</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">Ffg</font> =  <a href="bilinear.html#K8" title="BILINEAR:func.8">FunctionalSAF</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Maroon" title="c5">g</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#K8" title="BILINEAR:func.8">FunctionalSAF</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">w</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c9">Fg</font> =  <a href="bilinear.html#K8" title="BILINEAR:func.8">FunctionalSAF</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">w</font>);<br/>
<span class="kw">let </span><font color="Maroon" title="c10">v</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="hahnban1.html#D8" onmouseover="rs('hahnban1/D8')" onmouseout="rh()">HAHNBAN1:def 8</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">  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font> holds  <span class="p1">(<span class="default"><a href="bilinear.html#K8" title="BILINEAR:func.8">FunctionalSAF</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Maroon" title="c5">g</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="Olive">b<sub>1</sub></font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c10">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></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#K8" title="BILINEAR:func.8">FunctionalSAF</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Maroon" title="c5">g</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">v</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c11">a</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">Scalar</a> of ; <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#K8" title="BILINEAR:func.8">FunctionalSAF</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Maroon" title="c5">g</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="c11">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c10">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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#K8" title="BILINEAR:func.8">FunctionalSAF</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Maroon" title="c5">g</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">v</font></span>)</span></span><br/>



<span class="kw">thus </span><span class="p1">(<span class="default"><a href="bilinear.html#K8" title="BILINEAR:func.8">FunctionalSAF</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Maroon" title="c5">g</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="c11">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c10">v</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K8" title="BILINEAR:func.8">FunctionalSAF</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">w</font>)</span>)</span> <a href="hahnban1.html#K3" title="HAHNBAN1:func.3">+</a> <span class="p2">(<span class="default"><a href="bilinear.html#K8" title="BILINEAR:func.8">FunctionalSAF</a> (<font color="Maroon" title="c5">g</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="c11">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c10">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T12" target="_self" onmouseover="rs('bilinear/T12')" onmouseout="rh()">Th12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K8" title="BILINEAR:func.8">FunctionalSAF</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="c11">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c10">v</font></span>)</span></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#K8" title="BILINEAR:func.8">FunctionalSAF</a> (<font color="Maroon" title="c5">g</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="c11">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c10">v</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D3" onmouseover="rs('hahnban1/D3')" onmouseout="rh()">HAHNBAN1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">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#K8" title="BILINEAR:func.8">FunctionalSAF</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">v</font></span>)</span></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#K8" title="BILINEAR:func.8">FunctionalSAF</a> (<font color="Maroon" title="c5">g</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="c11">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c10">v</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D8" onmouseover="rs('hahnban1/D8')" onmouseout="rh()">HAHNBAN1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">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#K8" title="BILINEAR:func.8">FunctionalSAF</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">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="c11">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#K8" title="BILINEAR:func.8">FunctionalSAF</a> (<font color="Maroon" title="c5">g</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">v</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D8" onmouseover="rs('hahnban1/D8')" onmouseout="rh()">HAHNBAN1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c11">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#K8" title="BILINEAR:func.8">FunctionalSAF</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">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#K8" title="BILINEAR:func.8">FunctionalSAF</a> (<font color="Maroon" title="c5">g</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">v</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/>.= 
<font color="Maroon" title="c11">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#K8" title="BILINEAR:func.8">FunctionalSAF</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">w</font>)</span>)</span> <a href="hahnban1.html#K3" title="HAHNBAN1:func.3">+</a> <span class="p3">(<span class="default"><a href="bilinear.html#K8" title="BILINEAR:func.8">FunctionalSAF</a> (<font color="Maroon" title="c5">g</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">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D3" onmouseover="rs('hahnban1/D3')" onmouseout="rh()">HAHNBAN1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c11">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#K8" title="BILINEAR:func.8">FunctionalSAF</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Maroon" title="c5">g</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">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T12" target="_self" onmouseover="rs('bilinear/T12')" onmouseout="rh()">Th12</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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