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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">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">a'</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>  st not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b4">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">b'</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">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> holds <br/> ex <font color="Olive" title="b6">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">a'</font>,<font color="Olive" title="b6">c'</font> &amp; <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">b'</font>,<font color="Olive" title="b6">c'</font> )</span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">a'</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"> ( not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> &amp; <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> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">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> implies  ex <font color="Olive" title="b1">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a'</font>,<font color="Olive" title="b1">c'</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b'</font>,<font color="Olive" title="b1">c'</font> ) )</span><br/>











<b>assume </b><a NAME="E1:71"/><i><font color="Green" title="E58">A1</font></i>: 
( not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> &amp; <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> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">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> )
 ; <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">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a'</font>,<font color="Olive" title="b1">c'</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b'</font>,<font color="Olive" title="b1">c'</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_1"/><i><font color="Green" title="E59">A2</font></i>: 
<font color="Maroon" title="c1">AS</font> is  not   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</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">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a'</font>,<font color="Olive" title="b1">c'</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b'</font>,<font color="Olive" title="b1">c'</font> )</span><br/><b>consider </b><font color="Maroon" title="c7">X</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="E3:71_1"/><i><font color="Green" title="E60">A3</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> )
 <b>and </b><br/><a NAME="E4:71_1"/><i><font color="Green" title="E61">A4</font></i>: 
<font color="Maroon" title="c7">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a> 
 <b>by </b><i><a class="ref" href="aff_4.html#T37" target="_self" title="AFF_4:th.37">Th37</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_1_1"/><i><font color="Green" title="E62">A5</font></i>: 
not <font color="Maroon" title="c5">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">  ex <font color="Olive" title="b1">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a'</font>,<font color="Olive" title="b1">c'</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b'</font>,<font color="Olive" title="b1">c'</font> )</span><br/><a NAME="E2:71_1_1"/><i><font color="Green" title="E63">A6</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:71"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/><b>set </b><font color="Maroon" title="c8">AB</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>;<br/><b>set </b><font color="Maroon" title="c9">AB'</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="E3:71_1_1"/><i><font color="Green" title="E64">A7</font></i>: 
( <font color="Maroon" title="c2">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">a</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c3">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">a</font>,<font color="Maroon" title="c3">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> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">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;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</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_4.html#E1:71"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="aff_4.html#E2:71_1_1"><i><font color="Green" title="E63">A6</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#T51" title="AFF_1:th.51">AFF_1:51</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/><a NAME="E4:71_1_1"/><i><font color="Green" title="E65">A8</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">X</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:71_1"><i><font color="Green" title="E60">A3</font></i></a>, <a class="txt" href="aff_4.html#E4:71_1"><i><font color="Green" title="E61">A4</font></i></a>, <a class="txt" href="aff_4.html#E2:71_1_1"><i><font color="Green" title="E63">A6</font></i></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></i>;<br/><a NAME="E5:71_1_1"/><i><font color="Green" title="E66">A9</font></i>: 
not <font color="Maroon" title="c6">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_1_1_1"/>
<font color="Maroon" title="c6">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="E2:71_1_1_1"/><b>then </b>
 <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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">X</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E4:71_1"><i><font color="Green" title="E61">A4</font></i></a>, <a class="txt" href="aff_4.html#E3:71_1_1"><i><font color="Green" title="E64">A7</font></i></a>, <a class="txt" href="aff_4.html#E4:71_1_1"><i><font color="Green" title="E65">A8</font></i></a>, <a class="ref" href="aff_4.html#T23" target="_self" title="AFF_4:th.23">Th23</a></i>;<br/>
<b>hence </b><a NAME="E3:71_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E1:71_1_1"><i><font color="Green" title="E62">A5</font></i></a>, <a class="txt" href="aff_4.html#E3:71_1_1"><i><font color="Green" title="E64">A7</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>consider </b><font color="Maroon" title="c10">Y</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="E7:71_1_1"/><i><font color="Green" title="E67">A10</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Y</font> &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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Y</font> )
 <b>and </b><br/><a NAME="E8:71_1_1"/><i><font color="Green" title="E68">A11</font></i>: 
<font color="Maroon" title="c10">Y</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a> 
 <b>by </b><i><a class="txt" href="aff_4.html#E3:71_1_1"><i><font color="Green" title="E64">A7</font></i></a>, <a class="ref" href="aff_4.html#T39" target="_self" title="AFF_4:th.39">Th39</a></i>;<br/><b>set </b><font color="Maroon" title="c11">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font>;<br/><b>set </b><font color="Maroon" title="c12">P</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">b'</font>;<br/><a NAME="E9:71_1_1"/><i><font color="Green" title="E69">A12</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> &amp; <font color="Maroon" title="c5">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> &amp; <font color="Maroon" title="c6">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E3:71_1_1"><i><font color="Green" title="E64">A7</font></i></a>, <a class="txt" href="aff_4.html#E7:71_1_1"><i><font color="Green" title="E67">A10</font></i></a></i>;<br/><a NAME="E10:71_1_1"/><i><font color="Green" title="E70">A13</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">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="c3">b</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">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">a</font>,<font color="Maroon" title="c5">a'</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="c2">a</font>,<font color="Maroon" title="c5">a'</font> &amp; <font color="Maroon" title="c3">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="c3">b</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="c3">b</font>,<font color="Maroon" title="c6">b'</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E3:71_1"><i><font color="Green" title="E60">A3</font></i></a>, <a class="txt" href="aff_4.html#E1:71_1_1"><i><font color="Green" title="E62">A5</font></i></a>, <a class="txt" href="aff_4.html#E5:71_1_1"><i><font color="Green" title="E66">A9</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/><a NAME="E11:71_1_1"/><i><font color="Green" title="E71">A14</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Y</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">b'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Y</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E3:71_1"><i><font color="Green" title="E60">A3</font></i></a>, <a class="txt" href="aff_4.html#E1:71_1_1"><i><font color="Green" title="E62">A5</font></i></a>, <a class="txt" href="aff_4.html#E3:71_1_1"><i><font color="Green" title="E64">A7</font></i></a>, <a class="txt" href="aff_4.html#E5:71_1_1"><i><font color="Green" title="E66">A9</font></i></a>, <a class="txt" href="aff_4.html#E7:71_1_1"><i><font color="Green" title="E67">A10</font></i></a>, <a class="txt" href="aff_4.html#E8:71_1_1"><i><font color="Green" title="E68">A11</font></i></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></i>;<br/><a NAME="E12:71_1_1"/><i><font color="Green" title="E72">A15</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">b'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_1_1_2"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">b</font>,<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="E2:71_1_1_2"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E2:71_1_1"><i><font color="Green" title="E63">A6</font></i></a>, <a class="txt" href="aff_4.html#E3:71_1_1"><i><font color="Green" title="E64">A7</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green" title="E70">A13</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="E3:71_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E1:71_1_1"><i><font color="Green" title="E62">A5</font></i></a>, <a class="txt" href="aff_4.html#E4:71_1_1"><i><font color="Green" title="E65">A8</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green" title="E70">A13</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><div><i><font color="Green" title="E73">A16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_1_1_3"/><i><font color="Green" title="E74">A17</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">b</font>,<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">  ex <font color="Olive" title="b1">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a'</font>,<font color="Olive" title="b1">c'</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b'</font>,<font color="Olive" title="b1">c'</font> )</span><br/><b>set </b><font color="Maroon" title="c13">C</font> = <font color="Maroon" title="c4">c</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font></span>)</span>;<br/><a NAME="E2:71_1_1_3"/><i><font color="Green" title="E75">A18</font></i>: 
( <font color="Maroon" title="c4">c</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font></span>)</span> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font></span>)</span> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">c</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font></span>)</span> )
 <b>by </b><i><a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green" title="E70">A13</font></i></a>, <a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</a>, <a class="ref" href="aff_4.html#T27" target="_self" title="AFF_4:th.27">Th27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">c'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E4:71_1_1_3"/><i><font color="Green" title="E76">A19</font></i>: 
( <font color="Maroon" title="c14">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font></span>)</span> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c14">c'</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green" title="E70">A13</font></i></a>, <a class="txt" href="aff_4.html#E9"><i><font color="Green" title="E7">Lm2</font></i></a></i>;<br/><a NAME="E5:71_1_1_3"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_1_1_3_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:71_1_1_3_1"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E2:71_1_1"><i><font color="Green" title="E63">A6</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green" title="E70">A13</font></i></a>, <a class="txt" href="aff_4.html#E2:71_1_1_3"><i><font color="Green" title="E75">A18</font></i></a>, <a class="ref" href="aff_1.html#T71" title="AFF_1:th.71">AFF_1:71</a></i>;<br/>
<a NAME="E3:71_1_1_3_1"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">X</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E3:71_1"><i><font color="Green" title="E60">A3</font></i></a>, <a class="txt" href="aff_4.html#E4:71_1"><i><font color="Green" title="E61">A4</font></i></a>, <a class="txt" href="aff_4.html#E2:71_1_1"><i><font color="Green" title="E63">A6</font></i></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></i>;<br/>
<b>hence </b><a NAME="E4:71_1_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E1:71_1_1"><i><font color="Green" title="E62">A5</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green" title="E70">A13</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="E6:71_1_1_3"/><b>then </b>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c14">c'</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:71"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="aff_4.html#E1:71_1"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green" title="E70">A13</font></i></a>, <a class="txt" href="aff_4.html#E12:71_1_1"><i><font color="Green" title="E72">A15</font></i></a>, <a class="txt" href="aff_4.html#E1:71_1_1_3"><i><font color="Green" title="E74">A17</font></i></a>, <a class="txt" href="aff_4.html#E2:71_1_1_3"><i><font color="Green" title="E75">A18</font></i></a>, <a class="txt" href="aff_4.html#E4:71_1_1_3"><i><font color="Green" title="E76">A19</font></i></a>, <a class="ref" href="aff_4.html#T51" target="_self" title="AFF_4:th.51">Th51</a></i>;<br/><b>hence </b><a NAME="E7:71_1_1_3"/>
 ex <font color="Olive" title="b1">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a'</font>,<font color="Olive" title="b1">c'</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b'</font>,<font color="Olive" title="b1">c'</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E4:71_1_1_3"><i><font color="Green" title="E76">A19</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_4"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c13">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><a NAME="E2:71_1_1_4"/><i><font color="Green" title="E74">A20</font></i>: 
( <font color="Maroon" title="c13">q</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">a</font>,<font color="Maroon" title="c5">a'</font> &amp; <font color="Maroon" title="c13">q</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="c3">b</font>,<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">  ex <font color="Olive" title="b1">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a'</font>,<font color="Olive" title="b1">c'</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b'</font>,<font color="Olive" title="b1">c'</font> )</span><br/><a NAME="E3:71_1_1_4"/><i><font color="Green" title="E75">A21</font></i>: 
<font color="Maroon" title="c13">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_1_1_4_1"/>
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</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="E2:71_1_1_4_1"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">b'</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E2:71_1_1"><i><font color="Green" title="E63">A6</font></i></a>, <a class="txt" href="aff_4.html#E3:71_1_1"><i><font color="Green" title="E64">A7</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green" title="E70">A13</font></i></a>, <a class="txt" href="aff_4.html#E2:71_1_1_4"><i><font color="Green" title="E74">A20</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="E3:71_1_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E4:71_1_1"><i><font color="Green" title="E65">A8</font></i></a>, <a class="txt" href="aff_4.html#E5:71_1_1"><i><font color="Green" title="E66">A9</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green" title="E70">A13</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="E4:71_1_1_4"/><i><font color="Green" title="E76">A22</font></i>: 
<font color="Maroon" title="c13">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_4_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_1_1_4_2"/>
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="E2:71_1_1_4_2"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E2:71_1_1"><i><font color="Green" title="E63">A6</font></i></a>, <a class="txt" href="aff_4.html#E3:71_1_1"><i><font color="Green" title="E64">A7</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green" title="E70">A13</font></i></a>, <a class="txt" href="aff_4.html#E2:71_1_1_4"><i><font color="Green" title="E74">A20</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="E3:71_1_1_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E1:71_1_1"><i><font color="Green" title="E62">A5</font></i></a>, <a class="txt" href="aff_4.html#E4:71_1_1"><i><font color="Green" title="E65">A8</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green" title="E70">A13</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="E5:71_1_1_4"/><i><font color="Green" title="E77">A23</font></i>: 
<font color="Maroon" title="c13">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:71"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="aff_4.html#E3:71_1"><i><font color="Green" title="E60">A3</font></i></a>, <a class="txt" href="aff_4.html#E4:71_1"><i><font color="Green" title="E61">A4</font></i></a>, <a class="txt" href="aff_4.html#E1:71_1_1"><i><font color="Green" title="E62">A5</font></i></a>, <a class="txt" href="aff_4.html#E8:71_1_1"><i><font color="Green" title="E68">A11</font></i></a>, <a class="txt" href="aff_4.html#E9:71_1_1"><i><font color="Green" title="E69">A12</font></i></a>, <a class="txt" href="aff_4.html#E11:71_1_1"><i><font color="Green" title="E71">A14</font></i></a>, <a class="txt" href="aff_4.html#E2:71_1_1_4"><i><font color="Green" title="E74">A20</font></i></a>, <a class="ref" href="aff_4.html#T25" target="_self" title="AFF_4:th.25">Th25</a></i>;<br/><b>set </b><font color="Maroon" title="c14">C</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c4">c</font>;<br/><a NAME="E6:71_1_1_4"/><i><font color="Green" title="E78">A24</font></i>: 
( <font color="Maroon" title="c13">q</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="c13">q</font>,<font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c4">c</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="c13">q</font>,<font color="Maroon" title="c4">c</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c4">c</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>by </b><i><a class="txt" href="aff_4.html#E5:71_1_1_4"><i><font color="Green" title="E77">A23</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/><a NAME="E7:71_1_1_4"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green" title="E70">A13</font></i></a>, <a class="txt" href="aff_4.html#E2:71_1_1_4"><i><font color="Green" title="E74">A20</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">c'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E9:71_1_1_4"/><i><font color="Green" title="E79">A25</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c15">c'</font>
 <b>and </b><br/><a NAME="E10:71_1_1_4"/><i><font color="Green" title="E80">A26</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c15">c'</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:71_1_1_4"><i><font color="Green" title="E75">A21</font></i></a>, <a class="ref" href="aff_4.html#T1" target="_self" title="AFF_4:th.1">Th1</a></i>;<br/><a NAME="E11:71_1_1_4"/><i><font color="Green" title="E81">A27</font></i>: 
<font color="Maroon" title="c15">c'</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="c13">q</font>,<font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E5:71_1_1_4"><i><font color="Green" title="E77">A23</font></i></a>, <a class="txt" href="aff_4.html#E6:71_1_1_4"><i><font color="Green" title="E78">A24</font></i></a>, <a class="txt" href="aff_4.html#E9:71_1_1_4"><i><font color="Green" title="E79">A25</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/><a NAME="E12:71_1_1_4"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c4">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_4_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_1_1_4_3"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c4">c</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="E2:71_1_1_4_3"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E2:71_1_1"><i><font color="Green" title="E63">A6</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green" title="E70">A13</font></i></a>, <a class="txt" href="aff_4.html#E6:71_1_1_4"><i><font color="Green" title="E78">A24</font></i></a>, <a class="ref" href="aff_1.html#T71" title="AFF_1:th.71">AFF_1:71</a></i>;<br/>
<a NAME="E3:71_1_1_4_3"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">X</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E3:71_1"><i><font color="Green" title="E60">A3</font></i></a>, <a class="txt" href="aff_4.html#E4:71_1"><i><font color="Green" title="E61">A4</font></i></a>, <a class="txt" href="aff_4.html#E2:71_1_1"><i><font color="Green" title="E63">A6</font></i></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></i>;<br/>
<b>hence </b><a NAME="E4:71_1_1_4_3"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E1:71_1_1"><i><font color="Green" title="E62">A5</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green" title="E70">A13</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="E13:71_1_1_4"/><b>then </b>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c15">c'</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:71"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="aff_4.html#E1:71_1"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green" title="E70">A13</font></i></a>, <a class="txt" href="aff_4.html#E12:71_1_1"><i><font color="Green" title="E72">A15</font></i></a>, <a class="txt" href="aff_4.html#E2:71_1_1_4"><i><font color="Green" title="E74">A20</font></i></a>, <a class="txt" href="aff_4.html#E3:71_1_1_4"><i><font color="Green" title="E75">A21</font></i></a>, <a class="txt" href="aff_4.html#E4:71_1_1_4"><i><font color="Green" title="E76">A22</font></i></a>, <a class="txt" href="aff_4.html#E5:71_1_1_4"><i><font color="Green" title="E77">A23</font></i></a>, <a class="txt" href="aff_4.html#E6:71_1_1_4"><i><font color="Green" title="E78">A24</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1_4"><i><font color="Green" title="E80">A26</font></i></a>, <a class="txt" href="aff_4.html#E11:71_1_1_4"><i><font color="Green" title="E81">A27</font></i></a>, <a class="ref" href="aff_4.html#T49" target="_self" title="AFF_4:th.49">Th49</a></i>;<br/><b>hence </b><a NAME="E14:71_1_1_4"/>
 ex <font color="Olive" title="b1">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a'</font>,<font color="Olive" title="b1">c'</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b'</font>,<font color="Olive" title="b1">c'</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E10:71_1_1_4"><i><font color="Green" title="E80">A26</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="E15:71_1_1"/>
 ex <font color="Olive" title="b1">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a'</font>,<font color="Olive" title="b1">c'</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b'</font>,<font color="Olive" title="b1">c'</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E8:71_1_1"><i><font color="Green" title="E68">A11</font></i></a>, <a class="txt" href="aff_4.html#E10:71_1_1"><i><font color="Green" title="E70">A13</font></i></a>, <a class="txt" href="aff_4.html#E11:71_1_1"><i><font color="Green" title="E71">A14</font></i></a>, <a class="txt" href="aff_4.html#E13:71_1_1"><i><font color="Green" title="E73">A16</font></i></a>, <a class="ref" href="aff_4.html#T22" target="_self" title="AFF_4:th.22">Th22</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="E6:71_1"/>
 ex <font color="Olive" title="b1">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a'</font>,<font color="Olive" title="b1">c'</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b'</font>,<font color="Olive" title="b1">c'</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:71"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="aff_4.html#E3:71_1"><i><font color="Green" title="E60">A3</font></i></a>, <a class="txt" href="aff_4.html#E4:71_1"><i><font color="Green" title="E61">A4</font></i></a>, <a class="txt" href="aff_4.html#E70"><i><font color="Green" title="E56">Lm12</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="E3:71"/>
 ex <font color="Olive" title="b1">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a'</font>,<font color="Olive" title="b1">c'</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b'</font>,<font color="Olive" title="b1">c'</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:71"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="aff_4.html#T53" target="_self" title="AFF_4:th.53">Th53</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>
