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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">a</font>, <font color="Olive" title="b3">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b2">a</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="Olive" title="b1">S</font> &amp; <font color="Olive" title="b3">b</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="Olive" title="b1">S</font> holds <br/> ex <font color="Olive" title="b4">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">S</font> st <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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font>  st <font color="Olive" title="b1">a</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="c5">S</font> &amp; <font color="Olive" title="b2">b</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="c5">S</font> holds <br/> ex <font color="Olive" title="b3">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font> st <br/>( not <font color="Olive" title="b1">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Olive" title="b2">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )</span><br/><b>let </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">a</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="c5">S</font> &amp; <font color="Maroon" title="c7">b</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="c5">S</font> implies  ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font> st <br/>( not <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Maroon" title="c7">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  ) )</span><br/>







<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E16">A1</font></i>: 
( <font color="Maroon" title="c6">a</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="c5">S</font> &amp; <font color="Maroon" title="c7">b</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="c5">S</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font> st <br/>( not <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Maroon" title="c7">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )</span><br/>

<b>then consider </b><font color="Maroon" title="c8">a'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font><b> such that </b><br/><a NAME="E3:21"/><i><font color="Green" title="E17">A2</font></i>: 
not <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 <b>by </b><i><a class="ref" href="symsp_1.html#D1" target="_self" title="SYMSP_1:def.1">Def1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font> st <br/>( not <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Maroon" title="c7">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )</span><br/><b>consider </b><font color="Maroon" title="c9">b'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font><b> such that </b><br/><a NAME="E3:21_1"/><i><font color="Green" title="E19">A4</font></i>: 
not <font color="Maroon" title="c7">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 <b>by </b><i><a class="txt" href="symsp_1.html#E1:21"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="symsp_1.html#D1" target="_self" title="SYMSP_1:def.1">Def1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_1"/>
<font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font> st <br/>( not <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Maroon" title="c7">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )</span><br/><a NAME="E2:21_1_1"/><b>then </b>
( not <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Maroon" title="c7">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )
 <b>by </b><i><a class="txt" href="symsp_1.html#E3:21"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="symsp_1.html#E1:21_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="symsp_1.html#E3:21_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="symsp_1.html#T20" target="_self" title="SYMSP_1:th.20">Th20</a></i>;<br/><b>hence </b><a NAME="E3:21_1_1"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font> st <br/>( not <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Maroon" title="c7">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )
 ; <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="E5:21_1"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font> st <br/>( not <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Maroon" title="c7">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )
 <b>by </b><i><a class="txt" href="symsp_1.html#E3:21_1"><i><font color="Green" title="E19">A4</font></i></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>hence </b><a NAME="E5:21"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font> st <br/>( not <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Maroon" title="c7">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )
 <b>by </b><i><a class="txt" href="symsp_1.html#E3:21"><i><font color="Green" title="E17">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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