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<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">q</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">a'</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">b'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><br/> for <font color="Olive" title="b6">M</font>, <font color="Olive" title="b7">N</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">M</font> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b7">N</font> &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">M</font> &amp; <font color="Olive" title="b3">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">M</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b7">N</font> &amp; <font color="Olive" title="b5">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b7">N</font> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">b</font> &amp; <font color="Olive" title="b6">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">N</font> &amp; ( <font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">a'</font>,<font color="Olive" title="b5">b'</font> or <font color="Olive" title="b4">b</font>,<font color="Olive" title="b2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">b'</font>,<font color="Olive" title="b3">a'</font> ) &amp; <font color="Olive" title="b6">M</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b7">N</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a'</font> holds <br/><font color="Olive" title="b4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">b'</font></span><br/><b>let </b><font color="Maroon" title="c2">q</font>, <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">a'</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">b'</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"> for <font color="Olive" title="b1">M</font>, <font color="Olive" title="b2">N</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c4">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c6">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c2">q</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">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Olive" title="b1">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">N</font> &amp; ( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c6">b'</font> or <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c4">a'</font> ) &amp; <font color="Olive" title="b1">M</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b2">N</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a'</font> holds <br/><font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b'</font></span><br/><b>let </b><font color="Maroon" title="c7">M</font>, <font color="Maroon" title="c8">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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"> ( <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font> &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font> &amp; <font color="Maroon" title="c4">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> &amp; <font color="Maroon" title="c6">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> &amp; <font color="Maroon" title="c2">q</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">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c7">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">N</font> &amp; ( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c6">b'</font> or <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c4">a'</font> ) &amp; <font color="Maroon" title="c7">M</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c8">N</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a'</font> implies <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b'</font> )</span><br/>















<b>assume </b><b>that </b><br/><a NAME="E1:10"/><i><font color="Green" title="E10">A1</font></i>: 
( <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font> &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font> &amp; <font color="Maroon" title="c4">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> &amp; <font color="Maroon" title="c6">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> )
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green" title="E11">A2</font></i>: 
( <font color="Maroon" title="c2">q</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">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c7">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">N</font> )
 <b>and </b><br/><a NAME="E3:10"/><i><font color="Green" title="E12">A3</font></i>: 
( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c6">b'</font> or <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c4">a'</font> )
 <b>and </b><br/><a NAME="E4:10"/><i><font color="Green" title="E13">A4</font></i>: 
( <font color="Maroon" title="c7">M</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c8">N</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>and </b><br/><a NAME="E5:10"/><i><font color="Green" title="E14">A5</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b'</font></span><br/>

<a NAME="E6:10"/><i><font color="Green" title="E15">A6</font></i>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">a'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">b'</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E1:10"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="aff_4.html#E4:10"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E7:10"/><i><font color="Green" title="E16">A7</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c9">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E3:10_1"/><i><font color="Green" title="E17">A8</font></i>: 
( <font color="Maroon" title="c9">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E4:10_1"/>
( <font color="Maroon" title="c7">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c8">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E1:10"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="aff_4.html#E2:10"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="aff_4.html#E4:10"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="aff_4.html#E3:10_1"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E5:10_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E2:10"><i><font color="Green" title="E11">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/>


</div><b>end;</b></div>
<a NAME="E8:10"/><i><font color="Green" title="E17">A9</font></i>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c5">b</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E5:10"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="aff_1.html#T11" title="AFF_1:th.11">AFF_1:11</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E9:10"/>
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c6">b'</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E3:10"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<b>hence </b><a NAME="E10:10"/>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b'</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E6:10"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="aff_4.html#E7:10"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="aff_4.html#E8:10"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="aff_1.html#T70" title="AFF_1:th.70">AFF_1:70</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>
