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<span class="kw">let </span><font color="Maroon" title="c1">K</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="algstr_0.html#L2" title="ALGSTR_0:struct.2">addLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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>, <font color="Olive" title="b4">g</font>, <font color="Olive" title="b5">h</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> holds  <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Olive" title="b4">g</font></span>)</span> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Olive" title="b5">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">f</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">g</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Olive" title="b5">h</font></span>)</span></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</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> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Olive" title="b3">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">f</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Olive" title="b3">h</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font>, <font color="Maroon" title="c6">h</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Maroon" title="c6">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">f</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Maroon" title="c6">h</font></span>)</span></span><br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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  <span class="p1">(<span class="default"><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> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Maroon" title="c6">h</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">v</font>,<font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Maroon" title="c6">h</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">v</font>,<font color="Olive" title="b1">w</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><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> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Maroon" title="c6">h</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="c8">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Maroon" title="c6">h</font></span>)</span></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="c8">w</font></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><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> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Maroon" title="c6">h</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="c8">w</font> = 
<span class="p1">(<span class="default"><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> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c8">w</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="c6">h</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c8">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#D3" target="_self" title="BILINEAR:def.3">Def3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="c8">w</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c8">w</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="c6">h</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c8">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#D3" target="_self" title="BILINEAR:def.3">Def3</a></span>
<br/>.= 
<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="c8">w</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">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c8">w</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c8">w</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>
<br/>.= 
<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="c8">w</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">g</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Maroon" title="c6">h</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="c8">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#D3" target="_self" title="BILINEAR:def.3">Def3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Maroon" title="c6">h</font></span>)</span></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="c8">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#D3" target="_self" title="BILINEAR:def.3">Def3</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:13"/>
<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> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Maroon" title="c6">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">f</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="bilinear.html#K2" title="BILINEAR:func.2">+</a> <font color="Maroon" title="c6">h</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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