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<b>let </b><font color="Maroon">K</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="rlvect_1.html#L1">LoopStr</a> ;<br/><b>let </b><font color="Maroon">V</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of <font color="Maroon">K</font>, <font color="Maroon">W</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of <font color="Maroon">K</font>;<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">w</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="bilinear.html#NM1">Form</a> of <font color="Maroon">V</font>,<font color="Maroon">W</font>;<br/>









<b>set </b><font color="Maroon">F</font> =  <a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">f</font>,<font color="Maroon">y</font>;<br/>
<b>assume </b><a NAME="E1:75"/>
<font color="Maroon">f</font> is <a href="bilinear.html#V2" target="_self">additiveSAF</a>
 ;<br/>

<a NAME="E2:75"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">f</font>,<font color="Maroon">y</font> is <a href="hahnban1.html#V1">additive</a>
 
<b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span>,<font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K10" target="_self">FunctionalSAF</a> <font color="Maroon">f</font>,<font color="Maroon">y</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">w</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">y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</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">f</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></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"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K2">+</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">f</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span>
<b>by </b><i/>
;<br/>


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