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<span class="kw">let </span><font color="Maroon" title="c4">F</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="symsp_1.html#NM1" title="SYMSP_1:NM.1">SymSp</a> of <font color="Maroon" title="c4">F</font><br/> for <font color="Olive" title="b2">a9</font>, <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">b9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">S</font>  st not <font color="Olive" title="b3">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; <font color="Olive" title="b4">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Olive" title="b4">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; <font color="Olive" title="b3">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  holds <br/>( not <font color="Olive" title="b3">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Olive" title="b4">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">S</font> be   <a href="symsp_1.html#NM1" title="SYMSP_1:NM.1">SymSp</a> of <font color="Maroon" title="c4">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a9</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">b9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font>  st not <font color="Olive" title="b2">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; <font color="Olive" title="b3">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Olive" title="b3">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; <font color="Olive" title="b2">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  holds <br/>( not <font color="Olive" title="b2">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Olive" title="b3">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">a9</font>, <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">b9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c7">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; <font color="Maroon" title="c8">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Maroon" title="c8">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; <font color="Maroon" title="c7">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  implies ( not <font color="Maroon" title="c7">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Maroon" title="c8">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  ) )</span><br/>











<span class="kw">set </span><font color="Maroon" title="c10">0V</font> =  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">S</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:32"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
not <font color="Maroon" title="c7">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 <span class="kw">and </span><br/><a NAME="E2:32"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 <span class="kw">and </span><br/><a NAME="E3:32"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
not <font color="Maroon" title="c8">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 <span class="kw">and </span><br/><a NAME="E4:32"/><span class="lab"><font color="Green" title="E18">A4</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c7">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Maroon" title="c8">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E5:32"/>
( <font color="Maroon" title="c7">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  or <font color="Maroon" title="c8">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:32"/><span class="kw">then </span>
( ( <font color="Maroon" title="c7">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; <font color="Maroon" title="c7">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  ) or ( <font color="Maroon" title="c8">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; <font color="Maroon" title="c8">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="symsp_1.html#D1" target="_self" title="SYMSP_1:def.1">Def1</a></span>;<br/>
<a NAME="E7:32"/><span class="kw">then </span>
( ( <font color="Maroon" title="c7">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; <font color="Maroon" title="c7">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  ) or ( <font color="Maroon" title="c8">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; <font color="Maroon" title="c8">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T59" title="VECTSP_1:th.59">VECTSP_1:59</a></span>;<br/>
<a NAME="E8:32"/><span class="kw">then </span>
( <font color="Maroon" title="c7">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  or <font color="Maroon" title="c8">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="symsp_1.html#D1" target="_self" title="SYMSP_1:def.1">Def1</a></span>;<br/>
<a NAME="E9:32"/><span class="kw">then </span>
( <font color="Maroon" title="c7">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  or <font color="Maroon" title="c8">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>;<br/>
<a NAME="E10:32"/><span class="kw">then </span>
( <font color="Maroon" title="c7">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  or <font color="Maroon" title="c8">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:32"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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