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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#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="vectsp_1.html#V4">right-distributive</a>  <a href="vectsp_1.html#L3">doubleLoopStr</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">f</font> be  <a href="bilinear.html#V2" target="_self">additiveSAF</a> <a href="bilinear.html#V4" target="_self">homogeneousSAF</a> <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">V1</font> =  <a href="bilinear.html#K12" target="_self">leftker</a> <font color="Maroon">f</font>;<br/>
<b>thus </b><a NAME="E1:102"/>
for <font color="Olive">v</font>, <font color="Olive">u</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <font color="Maroon">f</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <font color="Maroon">f</font> holds <br/><font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <font color="Maroon">f</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_4.html#D1">VECTSP_4:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1"><b>proof </b></a><div class="add">

<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">u</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/>

<b>assume </b><a NAME="E1:102_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <font color="Maroon">f</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <font color="Maroon">f</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">v1</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:102_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; ( for <font color="Olive">w</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font> holds  <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v1</font>,<font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">u1</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E5:102_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; ( for <font color="Olive">w</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font> holds  <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">u1</font>,<font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> ) )
 <b>by </b><i><a class="ref" href="bilinear.html#D5" target="_self">Def5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">w</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font>;<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">u</font></span>)</span>,<font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v1</font>,<font color="Maroon">w</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">u1</font>,<font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#D8" target="_self">Def8</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</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">u1</font>,<font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:102_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <font color="Maroon">f</font>
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</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>;<br/>



<b>assume </b><a NAME="E2:102"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">v1</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:102"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; ( for <font color="Olive">w</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font> holds  <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v1</font>,<font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> ) )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">w</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font>;<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">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span>,<font color="Maroon">w</font> = 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v1</font>,<font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#D5" target="_self">Def5</a>, </i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#D5" target="_self">Def5</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:102"/>
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <font color="Maroon">f</font>
 ;<br/>


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