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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#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</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="hahnban1.html#NM1" title="HAHNBAN1:NM.1">Functional</a> of <font color="Olive" title="b1">V</font><br/> for <font color="Olive" title="b4">g</font> being  <a href="hahnban1.html#NM1" title="HAHNBAN1:NM.1">Functional</a> of <font color="Olive" title="b2">W</font>  st <font color="Olive" title="b3">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="hahnban1.html#K7" title="HAHNBAN1:func.7">0Functional</a> <font color="Olive" title="b1">V</font> holds <br/> <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K11" title="BILINEAR:func.11">FormFunctional</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Olive" title="b4">g</font></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="hahnban1.html#NM1" title="HAHNBAN1:NM.1">Functional</a> of <font color="Maroon" title="c2">V</font><br/> for <font color="Olive" title="b2">g</font> being  <a href="hahnban1.html#NM1" title="HAHNBAN1:NM.1">Functional</a> of <font color="Maroon" title="c3">W</font>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="hahnban1.html#K7" title="HAHNBAN1:func.7">0Functional</a> <font color="Maroon" title="c2">V</font> holds <br/> <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K11" title="BILINEAR:func.11">FormFunctional</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Olive" title="b2">g</font></span><br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="hahnban1.html#NM1" title="HAHNBAN1:NM.1">Functional</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"> for <font color="Olive" title="b1">g</font> being  <a href="hahnban1.html#NM1" title="HAHNBAN1:NM.1">Functional</a> of <font color="Maroon" title="c3">W</font>  st <font color="Maroon" title="c4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="hahnban1.html#K7" title="HAHNBAN1:func.7">0Functional</a> <font color="Maroon" title="c2">V</font> holds <br/> <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K11" title="BILINEAR:func.11">FormFunctional</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Olive" title="b1">g</font></span><br/><b>let </b><font color="Maroon" title="c5">g</font> be   <a href="hahnban1.html#NM1" title="HAHNBAN1:NM.1">Functional</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"> ( <font color="Maroon" title="c4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="hahnban1.html#K7" title="HAHNBAN1:func.7">0Functional</a> <font color="Maroon" title="c2">V</font> implies  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K11" title="BILINEAR:func.11">FormFunctional</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Maroon" title="c5">g</font> )</span><br/>







<b>set </b><font color="Maroon" title="c6">fg</font> =  <a href="bilinear.html#K11" title="BILINEAR:func.11">FormFunctional</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>;<br/>
<a NAME="E1:119"/><i><font color="Green" title="E56">A1</font></i>: 
 <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font> : <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="vectsp10.html#D9" title="VECTSP10:def.9">VECTSP10:def 9</a></i>;<br/>
<b>assume </b><a NAME="E2:119"/><i><font color="Green" title="E57">A2</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="hahnban1.html#K7" title="HAHNBAN1:func.7">0Functional</a> <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">  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K11" title="BILINEAR:func.11">FormFunctional</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Maroon" title="c5">g</font></span><br/>

<b>thus </b><a NAME="E3:119"/>
 <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K11" title="BILINEAR:func.11">FormFunctional</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Maroon" title="c5">g</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Maroon" title="c5">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K11" title="BILINEAR:func.11">FormFunctional</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K11" title="BILINEAR:func.11">FormFunctional</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Maroon" title="c5">g</font> )</span><br/>

<b>assume </b><a NAME="E1:119_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K11" title="BILINEAR:func.11">FormFunctional</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Maroon" title="c5">g</font></span><br/>

<b>then consider </b><font color="Maroon" title="c8">w</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font><b> such that </b><br/><a NAME="E3:119_1"/><i><font color="Green" title="E58">A3</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">w</font> &amp; ( for <font color="Olive" title="b1">v</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font> holds  <span class="p1">(<span class="default"><a href="bilinear.html#K11" title="BILINEAR:func.11">FormFunctional</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">v</font>,<font color="Maroon" title="c8">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> ) )
 ;<br/>
<b>assume </b><a NAME="E4:119_1"/>
not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Maroon" title="c5">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:119_1"/><b>then </b><i><font color="Green" title="E59">A4</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 
<b>by </b><i><a class="txt" href="bilinear.html#E1:119"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="bilinear.html#E3:119_1"><i><font color="Green" title="E58">A3</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">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"> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hahnban1.html#K7" title="HAHNBAN1:func.7">0Functional</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">v</font></span><br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">v</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">w</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K11" title="BILINEAR:func.11">FormFunctional</a> <font color="Maroon" title="c4">f</font>,<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="c9">v</font>,<font color="Maroon" title="c8">w</font>
<b>by </b><i><a class="ref" href="bilinear.html#D11" target="_self" title="BILINEAR:def.11">Def11</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
<b>by </b><i><a class="txt" href="bilinear.html#E3:119_1"><i><font color="Green" title="E58">A3</font></i></a></i>
;<br/><b>hence </b><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">v</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
<b>by </b><i><a class="txt" href="bilinear.html#E5:119_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hahnban1.html#K7" title="HAHNBAN1:func.7">0Functional</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">v</font>
<b>by </b><i><a class="ref" href="hahnban1.html#T22" title="HAHNBAN1:th.22">HAHNBAN1:22</a></i>
;<br/> <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="E7:119_1"/>
contradiction
 <b>by </b><i><a class="txt" href="bilinear.html#E2:119"><i><font color="Green" title="E57">A2</font></i></a>, <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/>


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

<b>thus </b><a NAME="E4:119"/>
 <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Maroon" title="c5">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K11" title="BILINEAR:func.11">FormFunctional</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="bilinear.html#T53" target="_self" title="BILINEAR:th.53">Th53</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>
