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<div class="add">

<b>let </b><font color="Maroon" title="c1">AS</font> be   <a href="diraf.html#NM1" title="DIRAF:NM.1">AffinSpace</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">o</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">a'</font>, <font color="Olive" title="b5">b'</font>, <font color="Olive" title="b6">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  st not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">a'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b5">b'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b6">x</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">a'</font>,<font color="Olive" title="b5">b'</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">a'</font>,<font color="Olive" title="b6">x</font> holds <br/><font color="Olive" title="b5">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">x</font></span><br/><b>let </b><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">a'</font>, <font color="Maroon" title="c6">b'</font>, <font color="Maroon" title="c7">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">a'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">b'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c7">x</font> implies <font color="Maroon" title="c6">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> )</span><br/>













<b>assume </b><a NAME="E1:66"/><i><font color="Green" title="E53">A1</font></i>: 
( not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">a'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">b'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c7">x</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">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font></span><br/>

<a NAME="E2:66"/><b>then </b><i><font color="Green" title="E54">A2</font></i>: 
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T16" target="_self" title="AFF_1:th.16">Th16</a></i>;<br/>
<b>assume </b><a NAME="E3:66"/><i><font color="Green" title="E55">A3</font></i>: 
<font color="Maroon" title="c6">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">x</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="E4:66"/><i><font color="Green" title="E56">A4</font></i>: 
<font color="Maroon" title="c5">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_1"/>
<font color="Maroon" title="c5">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b'</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="E2:66_1"/><b>then </b>
( <font color="Maroon" title="c5">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o</font> &amp; <font color="Maroon" title="c6">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o</font> )
 
<b>by </b><i><a class="txt" href="aff_1.html#E1:66"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="aff_1.html#T68" target="_self" title="AFF_1:th.68">Th68</a></i>;<br/>
<b>hence </b><a NAME="E3:66_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_1.html#E1:66"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="aff_1.html#E3:66"><i><font color="Green" title="E55">A3</font></i></a>, <a class="ref" href="aff_1.html#T69" target="_self" title="AFF_1:th.69">Th69</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>
<a NAME="E5:66"/><i><font color="Green" title="E57">A5</font></i>: 
<font color="Maroon" title="c5">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">o</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_2"/>
<font color="Maroon" title="c5">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o</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="E2:66_2"/><b>then </b>
( <font color="Maroon" title="c6">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o</font> )
 
<b>by </b><i><a class="txt" href="aff_1.html#E1:66"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="aff_1.html#T69" target="_self" title="AFF_1:th.69">Th69</a></i>;<br/>
<b>hence </b><a NAME="E3:66_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_1.html#E3:66"><i><font color="Green" title="E55">A3</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>
<a NAME="E6:66"/>
<font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c7">x</font>
 
<b>by </b><i><a class="txt" href="aff_1.html#E1:66"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="aff_1.html#E2:66"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="aff_1.html#T14" target="_self" title="AFF_1:th.14">Th14</a></i>;<br/>
<a NAME="E7:66"/><b>then </b><i><font color="Green" title="E58">A6</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c7">x</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" target="_self" title="AFF_1:def.1">Def1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>;<br/>
<b>set </b><font color="Maroon" title="c9">C</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font>;<br/>
<b>set </b><font color="Maroon" title="c10">P</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font>;<br/>
<a NAME="E8:66"/><i><font color="Green" title="E59">A7</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c5">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font> &amp; <font color="Maroon" title="c6">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font> )
 
<b>by </b><i><a class="txt" href="aff_1.html#E2:66"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="aff_1.html#E4:66"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="aff_1.html#D3" target="_self" title="AFF_1:def.3">Def3</a>, <a class="ref" href="aff_1.html#T26" target="_self" title="AFF_1:th.26">Th26</a></i>;<br/>
<a NAME="E9:66"/><b>then </b><i><font color="Green" title="E60">A8</font></i>: 
( <font color="Maroon" title="c5">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c6">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font> )
 
<b>by </b><i><a class="txt" href="aff_1.html#E1:66"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="aff_1.html#E2:66"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="aff_1.html#E4:66"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="aff_1.html#E7:66"><i><font color="Green" title="E58">A6</font></i></a>, <a class="ref" href="aff_1.html#T39" target="_self" title="AFF_1:th.39">Th39</a></i>;<br/>
<a NAME="E10:66"/><b>then </b>
<font color="Maroon" title="c5">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font>
 
<b>by </b><i><a class="txt" href="aff_1.html#E3:66"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="aff_1.html#E8:66"><i><font color="Green" title="E59">A7</font></i></a>, <a class="ref" href="aff_1.html#T30" target="_self" title="AFF_1:th.30">Th30</a></i>;<br/>
<a NAME="E11:66"/><b>then </b>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="txt" href="aff_1.html#E5:66"><i><font color="Green" title="E57">A5</font></i></a>, <a class="txt" href="aff_1.html#E8:66"><i><font color="Green" title="E59">A7</font></i></a>, <a class="txt" href="aff_1.html#E9:66"><i><font color="Green" title="E60">A8</font></i></a>, <a class="ref" href="aff_1.html#T30" target="_self" title="AFF_1:th.30">Th30</a></i>;<br/>
<b>hence </b><a NAME="E12:66"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_1.html#E1:66"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="aff_1.html#E8:66"><i><font color="Green" title="E59">A7</font></i></a>, <a class="ref" href="aff_1.html#T33" target="_self" title="AFF_1:th.33">Th33</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>
