<?xml version="1.0"?>
<div class="add">

<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="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="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="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="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">ModuleStr</a> over <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="b4">g</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="b2">W</font> holds <br/> <a href="bilinear.html#K10" title="BILINEAR:func.10">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">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="b3">f</font></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">ModuleStr</a> over <font color="Maroon" title="c1">K</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="b2">g</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="c3">W</font> holds <br/> <a href="bilinear.html#K10" title="BILINEAR:func.10">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">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="b1">f</font></span><br/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="Olive" title="b1">g</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="c3">W</font> holds <br/> <a href="bilinear.html#K10" title="BILINEAR:func.10">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">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="Maroon" title="c4">f</font></span><br/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">g</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="c3">W</font> implies  <a href="bilinear.html#K10" title="BILINEAR:func.10">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">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="c4">f</font> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c6">fg</font> =  <a href="bilinear.html#K9" title="BILINEAR:func.9">FormFunctional</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>);<br/>
<span class="kw">assume </span><a NAME="E1:115"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
<font color="Maroon" title="c5">g</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="c3">W</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="bilinear.html#K10" title="BILINEAR:func.10">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">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="c4">f</font></span><br/>

<a NAME="E2:115"/><span class="lab"><font color="Green" title="E53">A2</font></span>: 
 <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V</font> : <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#D9" onmouseover="rs('vectsp10/D9')" onmouseout="rh()">VECTSP10:def 9</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:115"/>
 <a href="bilinear.html#K10" title="BILINEAR:func.10">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">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="c4">f</font>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K10" title="BILINEAR:func.10">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FormFunctional</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>)</span>)</span></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="115_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K10" title="BILINEAR:func.10">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">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="c4">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:115_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K10" title="BILINEAR:func.10">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FormFunctional</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c4">f</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">v</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E3:115_1"/><span class="lab"><font color="Green" title="E54">A3</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v</font>
 <span class="kw">and </span><br/><a NAME="E4:115_1"/><span class="lab"><font color="Green" title="E55">A4</font></span>: 
 for <font color="Olive" title="b1">w</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c3">W</font> holds  <span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">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="c8">v</font>,<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>
 ;<br/>
<span class="kw">assume </span><a NAME="E5:115_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="c4">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:115_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A5</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">v</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E3:115_1"><span class="lab"><font color="Green" title="E54">A3</font></span></a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="115_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">w</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c3">W</font> holds  <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> <span class="p1">(<span class="default"><a href="hahnban1.html#K7" title="HAHNBAN1:func.7">0Functional</a> <font color="Maroon" title="c3">W</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">w</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c3">W</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">w</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="c3">W</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">w</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="c8">v</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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="c9">w</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">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="c8">v</font>,<font color="Maroon" title="c9">w</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#D10" target="_self" onmouseover="rs('bilinear/D10')" onmouseout="rh()">Def10</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:115_1"><span class="lab"><font color="Green" title="E55">A4</font></span></a></span>
;<br/><span class="kw">hence </span><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">w</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:115_1"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="ref" href="vectsp_1.html#T12" onmouseover="rs('vectsp_1/T12')" onmouseout="rh()">VECTSP_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="hahnban1.html#K7" title="HAHNBAN1:func.7">0Functional</a> <font color="Maroon" title="c3">W</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#T14" onmouseover="rs('hahnban1/T14')" onmouseout="rh()">HAHNBAN1:14</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:115_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="funct_2.html#T63" onmouseover="rs('funct_2/T63')" onmouseout="rh()">FUNCT_2:63</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E4:115"/>
 <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K10" title="BILINEAR:func.10">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FormFunctional</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T50" target="_self" onmouseover="rs('bilinear/T50')" onmouseout="rh()">Th50</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
