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<b>let </b><font color="Maroon">w</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="bilinear.html#D13" target="_self">BILINEAR:def 13</a><br/>

<b>set </b><font color="Maroon">Ffg</font> =  <a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="bilinear.html#K8" target="_self">+</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">w</font>;<br/>
<b>set </b><font color="Maroon">Ff</font> =  <a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">f</font>,<font color="Maroon">w</font>;<br/>
<b>set </b><font color="Maroon">Fg</font> =  <a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">g</font>,<font color="Maroon">w</font>;<br/>
<a NAME="E1:59_1_1"/><i><font color="Green">E33</font></i>: 
(  <a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">f</font>,<font color="Maroon">w</font> is <a href="hahnban1.html#V1">additive</a> &amp;  <a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">g</font>,<font color="Maroon">w</font> is <a href="hahnban1.html#V1">additive</a> )
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">v</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Maroon">y</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="hahnban1.html#D11">HAHNBAN1:def 11</a><br/>

<b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="bilinear.html#K8" target="_self">+</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span> <a href="hahnban1.html#K3">+</a> <span class="p2">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">g</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">g</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D6">HAHNBAN1:def 6</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" target="_self">FunctionalSAF</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">g</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="hahnban1.html#D11">HAHNBAN1:def 11</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" target="_self">FunctionalSAF</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">g</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">g</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="hahnban1.html#D11">HAHNBAN1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">g</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">g</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">g</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">g</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span> <a href="hahnban1.html#K3">+</a> <span class="p4">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">g</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">g</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D6">HAHNBAN1:def 6</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" target="_self">FunctionalSAF</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span> <a href="hahnban1.html#K3">+</a> <span class="p3">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">g</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">g</font>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</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" target="_self">FunctionalSAF</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span> <a href="hahnban1.html#K3">+</a> <span class="p3">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">g</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span> <a href="hahnban1.html#K3">+</a> <span class="p3">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">g</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D6">HAHNBAN1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="bilinear.html#K8" target="_self">+</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span> <a href="hahnban1.html#K3">+</a> <span class="p3">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">g</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="bilinear.html#K8" target="_self">+</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="bilinear.html#K8" target="_self">+</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">w</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i/>
;<br/>


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