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<b>let </b><font color="Maroon" title="c1">K</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font>, <font color="Olive" title="b2">W</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="vectsp_1.html#L1" title="VECTSP_1:struct.1">VectSpStr</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b3">f</font> being  <a href="bilinear.html#NM1" title="BILINEAR:NM.1">Form</a> of <font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">W</font><br/> for <font color="Olive" title="b4">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b5">w</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Olive" title="b2">W</font> holds   <a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="bilinear.html#K3" title="BILINEAR:func.3">*</a> <font color="Olive" title="b3">f</font></span>)</span>,<font color="Olive" title="b5">w</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b4">a</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <span class="p1">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b5">w</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">V</font>, <font color="Maroon" title="c3">W</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="vectsp_1.html#L1" title="VECTSP_1:struct.1">VectSpStr</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="bilinear.html#NM1" title="BILINEAR:NM.1">Form</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font><br/> for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b3">w</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font> holds   <a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="bilinear.html#K3" title="BILINEAR:func.3">*</a> <font color="Olive" title="b1">f</font></span>)</span>,<font color="Olive" title="b3">w</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">a</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <span class="p1">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">w</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="bilinear.html#NM1" title="BILINEAR:NM.1">Form</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">w</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font> holds   <a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="bilinear.html#K3" title="BILINEAR:func.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Olive" title="b2">w</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">a</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <span class="p1">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b2">w</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">w</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font> holds   <a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</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="Olive" title="b1">w</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">a</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <span class="p1">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">w</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c6">w</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</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> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">a</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <span class="p1">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">w</font></span>)</span></span><br/>









<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">v</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">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#K10" title="BILINEAR:func.10">FunctionalSAF</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> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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#K10" title="BILINEAR:func.10">FunctionalSAF</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="c7">v</font></span><br/><b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</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> <font color="Maroon" title="c7">v</font> = 
<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> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c6">w</font>
<b>by </b><i><a class="ref" href="bilinear.html#T10" target="_self" title="BILINEAR:th.10">Th10</a></i>
<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"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c6">w</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#D4" target="_self" title="BILINEAR:def.4">Def4</a></i>
<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#K10" title="BILINEAR:func.10">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="c7">v</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T10" target="_self" title="BILINEAR:th.10">Th10</a></i>
<br/>.= 
<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#K10" title="BILINEAR:func.10">FunctionalSAF</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="c7">v</font>
<b>by </b><i><a class="ref" href="hahnban1.html#D9" title="HAHNBAN1:def.9">HAHNBAN1:def 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="E2:27"/>
 <a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</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> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">a</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <span class="p1">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">w</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</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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