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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="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a> <a href="vectsp_1.html#V9">Field-like</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="hahnban1.html#NM1">Functional</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="hahnban1.html#NM1">Functional</a> of <font color="Maroon">W</font>;<br/>







<b>set </b><font color="Maroon">fg</font> =  <a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>;<br/>
<a NAME="E1:117"/><i><font color="Green">E33</font></i>: 
 <a href="vectsp10.html#K8">ker</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> : <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="vectsp10.html#D9">VECTSP10:def 9</a></i>;<br/>
<b>assume </b><a NAME="E2:117"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="hahnban1.html#K7">0Functional</a> <font color="Maroon">W</font>
 ;<br/>

<b>thus </b><a NAME="E3:117"/>
 <a href="bilinear.html#K12" target="_self">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="vectsp10.html#K8">ker</a> <font color="Maroon">f</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:117_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:117_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">x</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  <span class="p1">(<span class="default"><a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</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>assume </b><a NAME="E4:117_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K8">ker</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E5:117_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="bilinear.html#T13" target="_self">Th13</a>, <a class="ref" href="bilinear.html#T15" target="_self">Th15</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_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/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">w</font>
<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="bilinear.html#T15" target="_self">Th15</a></i>
;<br/><b>hence </b><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i><a class="ref" href="bilinear.html#T16" target="_self">Th16</a>, <a class="ref" href="vectsp_1.html#T44">VECTSP_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hahnban1.html#K7">0Functional</a> <font color="Maroon">W</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="hahnban1.html#T22">HAHNBAN1:22</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:117_1"/>
contradiction
 <b>by </b><i><a class="ref" href="bilinear.html#T14" target="_self">Th14</a>, <a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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

<b>thus </b><a NAME="E4:117"/>
 <a href="vectsp10.html#K8">ker</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="bilinear.html#K12" target="_self">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>
 <b>by </b><i/>;<br/>


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