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<b>consider </b><font color="Maroon">v</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/>
<b>consider </b><font color="Maroon">w</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E3:56_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="anproj_1.html#D8">ANPROJ_1:def 8</a></i>;<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/>
<b>assume </b><a NAME="E4:56_1_1"/><i><font color="Green">E37</font></i>: 
 <a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> is <a href="realset1.html#V1">trivial</a>
 ;<br/>

<a NAME="E5:56_1_1"/>
 <a href="relset_1.html#K4">dom</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="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:56_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>,<span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">W</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<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> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>,<span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">W</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<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></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E7:56_1_1"/><i><font color="Green">E81</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">W</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_1_1"/>
(  <a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st  <a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, <a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1"><b>suppose </b></a><a NAME="E1:56_1_1_1_1"/>
 <a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:56_1_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_2"><b>suppose </b></a><a NAME="E1:56_1_1_1_2"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st  <a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:56_1_1_1_2"/><i><font color="Green">E82</font></i>: 
 <a href="bilinear.html#K11" target="_self">FormFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E4:56_1_1_1_2"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>,<span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">W</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<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> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>,<span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">W</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<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></span>)</span></span><a href="domain_1.html#K1">]</a></span> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:56_1_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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