<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">AP</font> be   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a>  implies <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:39"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a> </span><br/>

<a NAME="E2:39"/><span class="kw">then </span>
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V13" title="AFF_2:attr.13">satisfying_pap</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_2.html#T23" target="_self" title="AFF_2:th.23">Th23</a></span>;<br/>
<a NAME="E3:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V1" title="AFF_2:attr.1">satisfying_PPAP</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="aff_2.html#T24" target="_self" title="AFF_2:th.24">Th24</a></span>;<br/>
<span class="kw">thus </span><a NAME="E4:39"/>
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="aff_2.html#D4" target="_self" title="AFF_2:def.4">AFF_2:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><br/> for <font color="Olive" title="b3">o</font>, <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font>, <font color="Olive" title="b6">c</font>, <font color="Olive" title="b7">a9</font>, <font color="Olive" title="b8">b9</font>, <font color="Olive" title="b9">c9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>  st <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">a</font> &amp; <font color="Olive" title="b3">o</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="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">c</font> &amp; <font color="Olive" title="b4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b7">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b9">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp; <font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b4">a</font>,<font color="Olive" title="b5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">a9</font>,<font color="Olive" title="b8">b9</font> &amp; <font color="Olive" title="b4">a</font>,<font color="Olive" title="b6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">a9</font>,<font color="Olive" title="b9">c9</font> holds <br/><font color="Olive" title="b5">b</font>,<font color="Olive" title="b6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b8">b9</font>,<font color="Olive" title="b9">c9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font>, <font color="Olive" title="b5">a9</font>, <font color="Olive" title="b6">b9</font>, <font color="Olive" title="b7">c9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>  st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b1">o</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">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">c</font> &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b5">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b6">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b7">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</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> &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="b5">a9</font>,<font color="Olive" title="b6">b9</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">a9</font>,<font color="Olive" title="b7">c9</font> holds <br/><font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">b9</font>,<font color="Olive" title="b7">c9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">o</font>, <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c9">a9</font>, <font color="Maroon" title="c10">b9</font>, <font color="Maroon" title="c11">c9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c10">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c11">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</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> &amp; <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> &amp; <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font> implies <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> )</span><br/>



















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:39_1"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E2:39_1"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E3:39_1"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E4:39_1"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">and </span><br/><a NAME="E5:39_1"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">and </span><br/><a NAME="E6:39_1"/>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c</font>
 <span class="kw">and </span><br/><a NAME="E7:39_1"/><span class="lab"><font color="Green" title="E25">A8</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E8:39_1"/><span class="lab"><font color="Green" title="E26">A9</font></span>: 
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E9:39_1"/><span class="lab"><font color="Green" title="E27">A10</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E10:39_1"/><span class="lab"><font color="Green" title="E28">A11</font></span>: 
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E11:39_1"/><span class="lab"><font color="Green" title="E29">A12</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E12:39_1"/><span class="lab"><font color="Green" title="E30">A13</font></span>: 
<font color="Maroon" title="c11">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E13:39_1"/><span class="lab"><font color="Green" title="E31">A14</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E14:39_1"/><span class="lab"><font color="Green" title="E32">A15</font></span>: 
<font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E15:39_1"/><span class="lab"><font color="Green" title="E33">A16</font></span>: 
<font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E16:39_1"/><span class="lab"><font color="Green" title="E34">A17</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E17:39_1"/><span class="lab"><font color="Green" title="E35">A18</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E18:39_1"/><span class="lab"><font color="Green" title="E36">A19</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font>
 <span class="kw">and </span><br/><a NAME="E19:39_1"/><span class="lab"><font color="Green" title="E37">A20</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font></span><br/>

<span class="kw">assume </span><a NAME="E20:39_1"/><span class="lab"><font color="Green" title="E38">A21</font></span>: 
not <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E21:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A22</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T12" title="AFF_1:th.12">AFF_1:12</a></span>;<br/>
<a NAME="E22:39_1"/><span class="lab"><font color="Green" title="E40">A23</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E7:39_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E11:39_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E15:39_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E17:39_1"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E23:39_1"/><span class="lab"><font color="Green" title="E41">A24</font></span>: 
not <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1_1"/><span class="lab"><font color="Green" title="E42">A25</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:39_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E5:39_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E9:39_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E10:39_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E14:39_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="txt" href="#E15:39_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:39_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:39_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E12:39_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="txt" href="#E15:39_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E20:39_1"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="txt" href="#E1:39_1_1"><span class="lab"><font color="Green" title="E42">A25</font></span></a>, <a class="ref" href="aff_1.html#T65" title="AFF_1:th.65">AFF_1:65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E24:39_1"/><span class="lab"><font color="Green" title="E42">A26</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E7:39_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E9:39_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E14:39_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="txt" href="#E16:39_1"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E25:39_1"/><span class="lab"><font color="Green" title="E43">A27</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1_2"/><span class="lab"><font color="Green" title="E44">A28</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:39_1_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:39_1"><span class="lab"><font color="Green" title="E37">A20</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E3:39_1_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E4:39_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A29</font></span>: 
( <font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c9</font> or <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:39_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E12:39_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="txt" href="#E15:39_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<a NAME="E5:39_1_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:39_1"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="txt" href="#E1:39_1_2"><span class="lab"><font color="Green" title="E44">A28</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E6:39_1_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E7:39_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b9</font> or <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E10:39_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E14:39_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:39_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E7:39_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E14:39_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="txt" href="#E15:39_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E16:39_1"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="txt" href="#E17:39_1"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="txt" href="#E20:39_1"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="txt" href="#E4:39_1_2"><span class="lab"><font color="Green" title="E45">A29</font></span></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#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c12">M</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">N</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">D</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>;<br/>
<a NAME="E26:39_1"/><span class="lab"><font color="Green" title="E44">A30</font></span>: 
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:39_1_3"/><span class="lab"><font color="Green" title="E45">A31</font></span>: 
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:39_1"><span class="lab"><font color="Green" title="E37">A20</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:39_1_3"/><span class="lab"><font color="Green" title="E46">A32</font></span>: 
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:39_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E10:39_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E14:39_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="txt" href="#E16:39_1"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E4:39_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> or <font color="Maroon" title="c9">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:39_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E12:39_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="txt" href="#E15:39_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E1:39_1_3"><span class="lab"><font color="Green" title="E45">A31</font></span></a>, <a class="ref" href="aff_1.html#T62" title="AFF_1:th.62">AFF_1:62</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:39_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E7:39_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E15:39_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E17:39_1"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="txt" href="#E20:39_1"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="txt" href="#E2:39_1_3"><span class="lab"><font color="Green" title="E46">A32</font></span></a>, <a class="ref" href="aff_1.html#T12" title="AFF_1:th.12">AFF_1:12</a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E27:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A33</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E28:39_1"/><span class="lab"><font color="Green" title="E46">A34</font></span>: 
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1_4"/>
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:39_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A35</font></span>: 
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E12:39_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E15:39_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E17:39_1"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E3:39_1_4"/>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:39_1"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E4:39_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E10:39_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E14:39_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="txt" href="#E26:39_1"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E2:39_1_4"><span class="lab"><font color="Green" title="E47">A35</font></span></a>, <a class="ref" href="aff_1.html#T62" title="AFF_1:th.62">AFF_1:62</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:39_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E7:39_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E14:39_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="txt" href="#E16:39_1"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E29:39_1"/><span class="lab"><font color="Green" title="E47">A36</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1_5"/><span class="lab"><font color="Green" title="E48">A37</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:39_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E3:39_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:39_1"><span class="lab"><font color="Green" title="E37">A20</font></span></a>, <a class="txt" href="#E28:39_1"><span class="lab"><font color="Green" title="E46">A34</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E4:39_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:39_1"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="txt" href="#E26:39_1"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E5:39_1_5"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E6:39_1_5"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E7:39_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E8:39_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E9:39_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A38</font></span>: 
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:39_1"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="txt" href="#E24:39_1"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E10:39_1_5"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c9">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_5"><span class="lab"><font color="Green" title="E48">A37</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E11:39_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c9">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E12:39_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:39_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:39_1"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="txt" href="#E26:39_1"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E9:39_1_5"><span class="lab"><font color="Green" title="E49">A38</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E30:39_1"/><span class="lab"><font color="Green" title="E48">A39</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1_6"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:39_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E3:39_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:39_1"><span class="lab"><font color="Green" title="E37">A20</font></span></a>, <a class="txt" href="#E22:39_1"><span class="lab"><font color="Green" title="E40">A23</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E4:39_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:39_1"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="txt" href="#E24:39_1"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:39_1_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:39_1"><span class="lab"><font color="Green" title="E47">A36</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E49">A40</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_7"><span class="kw">now </span></a><div class="add"><a NAME="E1:39_1_7"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c9">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E7:39_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><a NAME="E2:39_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><a NAME="E3:39_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A41</font></span>: 
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c15">M</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>;<br/><span class="kw">set </span><font color="Maroon" title="c16">N</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>;<br/><span class="kw">set </span><font color="Maroon" title="c17">D</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>;<br/><a NAME="E4:39_1_7"/><span class="lab"><font color="Green" title="E51">A42</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:39_1"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/><a NAME="E5:39_1_7"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:39_1"><span class="lab"><font color="Green" title="E39">A22</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">K</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E7:39_1_7"/><span class="lab"><font color="Green" title="E52">A43</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">K</font>
 <span class="kw">and </span><br/><a NAME="E8:39_1_7"/><span class="lab"><font color="Green" title="E53">A44</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c18">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></span>;<br/><a NAME="E9:39_1_7"/><span class="lab"><font color="Green" title="E54">A45</font></span>: 
<font color="Maroon" title="c18">K</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:39_1_7"><span class="lab"><font color="Green" title="E53">A44</font></span></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></span>;<br/><a NAME="E10:39_1_7"/><span class="lab"><font color="Green" title="E55">A46</font></span>: 
<font color="Maroon" title="c6">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="c6">a</font>,<font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:39_1"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/><a NAME="E11:39_1_7"/><span class="lab"><font color="Green" title="E56">A47</font></span>: 
<font color="Maroon" title="c7">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="c6">a</font>,<font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:39_1"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/><a NAME="E12:39_1_7"/><span class="lab"><font color="Green" title="E57">A48</font></span>: 
( <font color="Maroon" title="c7">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="c7">b</font>,<font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c8">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="c7">b</font>,<font color="Maroon" title="c8">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:39_1"><span class="lab"><font color="Green" title="E39">A22</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/><a NAME="E13:39_1_7"/>
not  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c18">K</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1_7_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c18">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:39_1_7_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">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="c7">b</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:39_1_7"><span class="lab"><font color="Green" title="E53">A44</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<a NAME="E3:39_1_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">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="c6">a</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:39_1_7"><span class="lab"><font color="Green" title="E57">A48</font></span></a>, <a class="txt" href="#E11:39_1_7"><span class="lab"><font color="Green" title="E56">A47</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:39_1_7_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:39_1"><span class="lab"><font color="Green" title="E48">A39</font></span></a>, <a class="txt" href="#E4:39_1_7"><span class="lab"><font color="Green" title="E51">A42</font></span></a>, <a class="txt" href="#E10:39_1_7"><span class="lab"><font color="Green" title="E55">A46</font></span></a>, <a class="txt" href="#E11:39_1_7"><span class="lab"><font color="Green" title="E56">A47</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E15:39_1_7"/><span class="lab"><font color="Green" title="E58">A49</font></span>: 
<font color="Maroon" title="c19">p</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="c6">a</font>,<font color="Maroon" title="c7">b</font>
 <span class="kw">and </span><br/><a NAME="E16:39_1_7"/><span class="lab"><font color="Green" title="E59">A50</font></span>: 
<font color="Maroon" title="c19">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_7"><span class="lab"><font color="Green" title="E51">A42</font></span></a>, <a class="txt" href="#E9:39_1_7"><span class="lab"><font color="Green" title="E54">A45</font></span></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></span>;<br/><a NAME="E17:39_1_7"/><span class="lab"><font color="Green" title="E60">A51</font></span>: 
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c5">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:39_1_7"><span class="lab"><font color="Green" title="E57">A48</font></span></a>, <a class="txt" href="#E7:39_1_7"><span class="lab"><font color="Green" title="E52">A43</font></span></a>, <a class="txt" href="#E8:39_1_7"><span class="lab"><font color="Green" title="E53">A44</font></span></a>, <a class="txt" href="#E16:39_1_7"><span class="lab"><font color="Green" title="E59">A50</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/><a NAME="E18:39_1_7"/><span class="lab"><font color="Green" title="E61">A52</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_7_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1_7_2"/>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:39_1_7_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_7"><span class="lab"><font color="Green" title="E51">A42</font></span></a>, <a class="txt" href="#E10:39_1_7"><span class="lab"><font color="Green" title="E55">A46</font></span></a>, <a class="txt" href="#E11:39_1_7"><span class="lab"><font color="Green" title="E56">A47</font></span></a>, <a class="txt" href="#E15:39_1_7"><span class="lab"><font color="Green" title="E58">A49</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/>
<a NAME="E3:39_1_7_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E7:39_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:39_1_7_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E5:39_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E9:39_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E14:39_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="txt" href="#E16:39_1"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">set </span><font color="Maroon" title="c20">R</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c19">p</font>;<br/><a NAME="E19:39_1_7"/><span class="lab"><font color="Green" title="E62">A53</font></span>: 
<font color="Maroon" title="c19">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_7_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1_7_3"/>
<font color="Maroon" title="c19">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:39_1_7_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E25:39_1"><span class="lab"><font color="Green" title="E43">A27</font></span></a>, <a class="txt" href="#E4:39_1_7"><span class="lab"><font color="Green" title="E51">A42</font></span></a>, <a class="txt" href="#E10:39_1_7"><span class="lab"><font color="Green" title="E55">A46</font></span></a>, <a class="txt" href="#E11:39_1_7"><span class="lab"><font color="Green" title="E56">A47</font></span></a>, <a class="txt" href="#E15:39_1_7"><span class="lab"><font color="Green" title="E58">A49</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:39_1_7_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E5:39_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E9:39_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E14:39_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="txt" href="#E16:39_1"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E20:39_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A54</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c19">p</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/><a NAME="E21:39_1_7"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:39_1"><span class="lab"><font color="Green" title="E40">A23</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">T</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E23:39_1_7"/><span class="lab"><font color="Green" title="E64">A55</font></span>: 
<font color="Maroon" title="c19">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">T</font>
 <span class="kw">and </span><br/><a NAME="E24:39_1_7"/><span class="lab"><font color="Green" title="E65">A56</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c21">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></span>;<br/><a NAME="E25:39_1_7"/><span class="lab"><font color="Green" title="E66">A57</font></span>: 
( <font color="Maroon" title="c6">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="c6">a</font>,<font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c8">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="c6">a</font>,<font color="Maroon" title="c8">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:39_1"><span class="lab"><font color="Green" title="E40">A23</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/><a NAME="E26:39_1_7"/><span class="lab"><font color="Green" title="E67">A58</font></span>: 
not <font color="Maroon" title="c4">C</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c21">T</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_7_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1_7_4"/>
<font color="Maroon" title="c4">C</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c21">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:39_1_7_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">C</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="c6">a</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:39_1_7"><span class="lab"><font color="Green" title="E65">A56</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<a NAME="E3:39_1_7_4"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:39_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E25:39_1_7"><span class="lab"><font color="Green" title="E66">A57</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:39_1_7_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E7:39_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E15:39_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E17:39_1"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E27:39_1_7"/>
<font color="Maroon" title="c21">T</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:39_1_7"><span class="lab"><font color="Green" title="E65">A56</font></span></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E29:39_1_7"/><span class="lab"><font color="Green" title="E68">A59</font></span>: 
<font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E30:39_1_7"/><span class="lab"><font color="Green" title="E69">A60</font></span>: 
<font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:39_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E26:39_1_7"><span class="lab"><font color="Green" title="E67">A58</font></span></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></span>;<br/><a NAME="E31:39_1_7"/><span class="lab"><font color="Green" title="E70">A61</font></span>: 
<font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c22">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:39_1_7"><span class="lab"><font color="Green" title="E66">A57</font></span></a>, <a class="txt" href="#E23:39_1_7"><span class="lab"><font color="Green" title="E64">A55</font></span></a>, <a class="txt" href="#E24:39_1_7"><span class="lab"><font color="Green" title="E65">A56</font></span></a>, <a class="txt" href="#E30:39_1_7"><span class="lab"><font color="Green" title="E69">A60</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/><a NAME="E32:39_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A62</font></span>: 
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c22">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E11:39_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E15:39_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E4:39_1_7"><span class="lab"><font color="Green" title="E51">A42</font></span></a>, <a class="txt" href="#E10:39_1_7"><span class="lab"><font color="Green" title="E55">A46</font></span></a>, <a class="txt" href="#E11:39_1_7"><span class="lab"><font color="Green" title="E56">A47</font></span></a>, <a class="txt" href="#E15:39_1_7"><span class="lab"><font color="Green" title="E58">A49</font></span></a>, <a class="txt" href="#E29:39_1_7"><span class="lab"><font color="Green" title="E68">A59</font></span></a>, <a class="txt" href="#E17:39_1_7"><span class="lab"><font color="Green" title="E60">A51</font></span></a>, <a class="ref" href="aff_2.html#D1" target="_self" title="AFF_2:def.1">Def1</a></span>;<br/><a NAME="E33:39_1_7"/><span class="lab"><font color="Green" title="E72">A63</font></span>: 
( <font color="Maroon" title="c9">a9</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="c9">a9</font>,<font color="Maroon" title="c19">p</font> &amp; <font color="Maroon" title="c19">p</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="c9">a9</font>,<font color="Maroon" title="c19">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:39_1_7"><span class="lab"><font color="Green" title="E62">A53</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/><span class="kw">assume </span><a NAME="E34:39_1_7"/>
not <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E35:39_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A64</font></span>: 
<font color="Maroon" title="c18">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:39_1_7"><span class="lab"><font color="Green" title="E57">A48</font></span></a>, <a class="txt" href="#E8:39_1_7"><span class="lab"><font color="Green" title="E53">A44</font></span></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></span>;<br/><a NAME="E36:39_1_7"/>
not <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c22">q</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c19">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_7_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1_7_5"/>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c22">q</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c19">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:39_1_7_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A65</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c19">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:39_1"><span class="lab"><font color="Green" title="E41">A24</font></span></a>, <a class="txt" href="#E29:39_1_7"><span class="lab"><font color="Green" title="E68">A59</font></span></a>, <a class="txt" href="#E32:39_1_7"><span class="lab"><font color="Green" title="E71">A62</font></span></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a></span>;<br/>
<a NAME="E3:39_1_7_5"/>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E7:39_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a>, <a class="ref" href="aff_1.html#T55" title="AFF_1:th.55">AFF_1:55</a></span>;<br/>
<a NAME="E4:39_1_7_5"/><span class="kw">then </span>
<font color="Maroon" title="c19">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E33:39_1_7"><span class="lab"><font color="Green" title="E72">A63</font></span></a>, <a class="txt" href="#E2:39_1_7_5"><span class="lab"><font color="Green" title="E74">A65</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a>, <a class="ref" href="aff_1.html#T67" title="AFF_1:th.67">AFF_1:67</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:39_1_7_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E7:39_1_7"><span class="lab"><font color="Green" title="E52">A43</font></span></a>, <a class="txt" href="#E9:39_1_7"><span class="lab"><font color="Green" title="E54">A45</font></span></a>, <a class="txt" href="#E16:39_1_7"><span class="lab"><font color="Green" title="E59">A50</font></span></a>, <a class="txt" href="#E18:39_1_7"><span class="lab"><font color="Green" title="E61">A52</font></span></a>, <a class="txt" href="#E35:39_1_7"><span class="lab"><font color="Green" title="E73">A64</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E38:39_1_7"/><span class="lab"><font color="Green" title="E74">A66</font></span>: 
<font color="Maroon" title="c23">r</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="c9">a9</font>,<font color="Maroon" title="c19">p</font>
 <span class="kw">and </span><br/><a NAME="E39:39_1_7"/><span class="lab"><font color="Green" title="E75">A67</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c23">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:39_1_7"><span class="lab"><font color="Green" title="E63">A54</font></span></a>, <a class="ref" href="aff_1.html#T73" title="AFF_1:th.73">AFF_1:73</a></span>;<br/><div><span class="lab"><font color="Green" title="E76">A68</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_7_6"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:39_1_7_6"/>
<font color="Maroon" title="c23">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">a9</font></span><br/><a NAME="E2:39_1_7_6"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c23">r</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E11:39_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E15:39_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E4:39_1_7"><span class="lab"><font color="Green" title="E51">A42</font></span></a>, <a class="txt" href="#E10:39_1_7"><span class="lab"><font color="Green" title="E55">A46</font></span></a>, <a class="txt" href="#E11:39_1_7"><span class="lab"><font color="Green" title="E56">A47</font></span></a>, <a class="txt" href="#E15:39_1_7"><span class="lab"><font color="Green" title="E58">A49</font></span></a>, <a class="txt" href="#E29:39_1_7"><span class="lab"><font color="Green" title="E68">A59</font></span></a>, <a class="txt" href="#E17:39_1_7"><span class="lab"><font color="Green" title="E60">A51</font></span></a>, <a class="txt" href="#E31:39_1_7"><span class="lab"><font color="Green" title="E70">A61</font></span></a>, <a class="ref" href="aff_2.html#D1" target="_self" title="AFF_2:def.1">Def1</a></span>;<br/><a NAME="E3:39_1_7_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A69</font></span>: 
<font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><a NAME="E4:39_1_7_6"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c9">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E7:39_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><a NAME="E5:39_1_7_6"/><span class="kw">then </span>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:39_1_7_6"/>
<font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E3:39_1_7_6"><span class="lab"><font color="Green" title="E77">A69</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E41:39_1_7"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:39_1_7"><span class="lab"><font color="Green" title="E75">A67</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E42:39_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c23">r</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><a NAME="E43:39_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c22">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c22">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><a NAME="E44:39_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c22">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:39_1"><span class="lab"><font color="Green" title="E41">A24</font></span></a>, <a class="txt" href="#E29:39_1_7"><span class="lab"><font color="Green" title="E68">A59</font></span></a>, <a class="txt" href="#E32:39_1_7"><span class="lab"><font color="Green" title="E71">A62</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/><a NAME="E45:39_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A70</font></span>: 
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c22">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E3:39_1_7"><span class="lab"><font color="Green" title="E50">A41</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/><a NAME="E46:39_1_7"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c19">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_7"><span class="lab"><font color="Green" title="E51">A42</font></span></a>, <a class="txt" href="#E10:39_1_7"><span class="lab"><font color="Green" title="E55">A46</font></span></a>, <a class="txt" href="#E11:39_1_7"><span class="lab"><font color="Green" title="E56">A47</font></span></a>, <a class="txt" href="#E15:39_1_7"><span class="lab"><font color="Green" title="E58">A49</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><a NAME="E47:39_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c19">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><a NAME="E48:39_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><a NAME="E49:39_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A71</font></span>: 
<font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:39_1"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="txt" href="#E24:39_1"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/><a NAME="E50:39_1_7"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c22">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:39_1_7"><span class="lab"><font color="Green" title="E75">A67</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E51:39_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c22">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><a NAME="E52:39_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:39_1_7"><span class="lab"><font color="Green" title="E77">A70</font></span></a>, <a class="txt" href="#E40:39_1_7"><span class="lab"><font color="Green" title="E76">A68</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/><a NAME="E53:39_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A72</font></span>: 
<font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c10">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E9:39_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E10:39_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E14:39_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="txt" href="#E20:39_1_7"><span class="lab"><font color="Green" title="E63">A54</font></span></a>, <a class="txt" href="#E33:39_1_7"><span class="lab"><font color="Green" title="E72">A63</font></span></a>, <a class="txt" href="#E38:39_1_7"><span class="lab"><font color="Green" title="E74">A66</font></span></a>, <a class="txt" href="#E49:39_1_7"><span class="lab"><font color="Green" title="E78">A71</font></span></a>, <a class="ref" href="aff_2.html#D1" target="_self" title="AFF_2:def.1">Def1</a></span>;<br/><a NAME="E54:39_1_7"/>
<font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c22">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:39_1"><span class="lab"><font color="Green" title="E37">A20</font></span></a>, <a class="txt" href="#E22:39_1"><span class="lab"><font color="Green" title="E40">A23</font></span></a>, <a class="txt" href="#E31:39_1_7"><span class="lab"><font color="Green" title="E70">A61</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/><a NAME="E55:39_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A73</font></span>: 
<font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c11">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E12:39_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="txt" href="#E15:39_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E29:39_1_7"><span class="lab"><font color="Green" title="E68">A59</font></span></a>, <a class="txt" href="#E20:39_1_7"><span class="lab"><font color="Green" title="E63">A54</font></span></a>, <a class="txt" href="#E33:39_1_7"><span class="lab"><font color="Green" title="E72">A63</font></span></a>, <a class="txt" href="#E38:39_1_7"><span class="lab"><font color="Green" title="E74">A66</font></span></a>, <a class="txt" href="#E45:39_1_7"><span class="lab"><font color="Green" title="E77">A70</font></span></a>, <a class="ref" href="aff_2.html#D1" target="_self" title="AFF_2:def.1">Def1</a></span>;<br/><a NAME="E56:39_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c10">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:39_1_7"><span class="lab"><font color="Green" title="E61">A52</font></span></a>, <a class="txt" href="#E53:39_1_7"><span class="lab"><font color="Green" title="E79">A72</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/><a NAME="E57:39_1_7"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c10">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><a NAME="E58:39_1_7"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c23">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E59:39_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c10">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c23">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><a NAME="E60:39_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A74</font></span>: 
<font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><a NAME="E61:39_1_7"/>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c11">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:39_1_7"><span class="lab"><font color="Green" title="E61">A52</font></span></a>, <a class="txt" href="#E17:39_1_7"><span class="lab"><font color="Green" title="E60">A51</font></span></a>, <a class="txt" href="#E55:39_1_7"><span class="lab"><font color="Green" title="E80">A73</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/><a NAME="E62:39_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c23">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:39_1"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="txt" href="#E60:39_1_7"><span class="lab"><font color="Green" title="E81">A74</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/><a NAME="E63:39_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:39_1_7"><span class="lab"><font color="Green" title="E79">A72</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E64:39_1_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:39_1"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="txt" href="#E18:39_1_7"><span class="lab"><font color="Green" title="E61">A52</font></span></a>, <a class="txt" href="#E17:39_1_7"><span class="lab"><font color="Green" title="E60">A51</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E32:39_1"/><span class="lab"><font color="Green" title="E50">A75</font></span>: 
<font color="Maroon" title="c10">b9</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="c9">a9</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:39_1"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E33:39_1"/><span class="lab"><font color="Green" title="E51">A76</font></span>: 
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:39_1"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="ref" href="aff_1.html#T12" title="AFF_1:th.12">AFF_1:12</a></span>;<br/>
<a NAME="E34:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A77</font></span>: 
( <font color="Maroon" title="c10">b9</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="c10">b9</font>,<font color="Maroon" title="c11">c9</font> &amp; <font color="Maroon" title="c11">c9</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="c10">b9</font>,<font color="Maroon" title="c11">c9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E35:39_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:39_1"><span class="lab"><font color="Green" title="E51">A76</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">K</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E37:39_1"/><span class="lab"><font color="Green" title="E53">A78</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">K</font>
 <span class="kw">and </span><br/><a NAME="E38:39_1"/><span class="lab"><font color="Green" title="E54">A79</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c15">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></span>;<br/>
<a NAME="E39:39_1"/><span class="lab"><font color="Green" title="E55">A80</font></span>: 
<font color="Maroon" title="c15">K</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:39_1"><span class="lab"><font color="Green" title="E54">A79</font></span></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></span>;<br/>
<a NAME="E40:39_1"/><span class="lab"><font color="Green" title="E56">A81</font></span>: 
<font color="Maroon" title="c9">a9</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="c9">a9</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:39_1"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E41:39_1"/>
not  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c15">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1_8"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c15">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:39_1_8"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</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="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:39_1"><span class="lab"><font color="Green" title="E54">A79</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<a NAME="E3:39_1_8"/><span class="kw">then </span>
<font color="Maroon" title="c11">c9</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="c9">a9</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:39_1"><span class="lab"><font color="Green" title="E52">A77</font></span></a>, <a class="txt" href="#E32:39_1"><span class="lab"><font color="Green" title="E50">A75</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:39_1_8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:39_1"><span class="lab"><font color="Green" title="E47">A36</font></span></a>, <a class="txt" href="#E27:39_1"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="txt" href="#E40:39_1"><span class="lab"><font color="Green" title="E56">A81</font></span></a>, <a class="txt" href="#E32:39_1"><span class="lab"><font color="Green" title="E50">A75</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E43:39_1"/><span class="lab"><font color="Green" title="E57">A82</font></span>: 
<font color="Maroon" title="c16">p</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="c9">a9</font>,<font color="Maroon" title="c10">b9</font>
 <span class="kw">and </span><br/><a NAME="E44:39_1"/><span class="lab"><font color="Green" title="E58">A83</font></span>: 
<font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:39_1"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="txt" href="#E39:39_1"><span class="lab"><font color="Green" title="E55">A80</font></span></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></span>;<br/>
<a NAME="E45:39_1"/><span class="lab"><font color="Green" title="E59">A84</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1_9"/><span class="lab"><font color="Green" title="E60">A85</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:39_1_9"/>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:39_1"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E3:39_1_9"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E9:39_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E10:39_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E14:39_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="txt" href="#E26:39_1"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E1:39_1_9"><span class="lab"><font color="Green" title="E60">A85</font></span></a>, <a class="ref" href="aff_1.html#T62" title="AFF_1:th.62">AFF_1:62</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:39_1_9"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E7:39_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E14:39_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="txt" href="#E16:39_1"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E46:39_1"/><span class="lab"><font color="Green" title="E60">A86</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1_10"/>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:39_1_10"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:39_1"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="txt" href="#E40:39_1"><span class="lab"><font color="Green" title="E56">A81</font></span></a>, <a class="txt" href="#E32:39_1"><span class="lab"><font color="Green" title="E50">A75</font></span></a>, <a class="txt" href="#E43:39_1"><span class="lab"><font color="Green" title="E57">A82</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/>
<a NAME="E3:39_1_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A87</font></span>: 
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E45:39_1"><span class="lab"><font color="Green" title="E59">A84</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<a NAME="E4:39_1_10"/>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:39_1"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E5:39_1_10"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E26:39_1"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E3:39_1_10"><span class="lab"><font color="Green" title="E61">A87</font></span></a>, <a class="ref" href="aff_1.html#T62" title="AFF_1:th.62">AFF_1:62</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:39_1_10"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E5:39_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E9:39_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E14:39_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="txt" href="#E16:39_1"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E47:39_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:39_1"><span class="lab"><font color="Green" title="E46">A34</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">T</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E49:39_1"/><span class="lab"><font color="Green" title="E61">A88</font></span>: 
<font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">T</font>
 <span class="kw">and </span><br/><a NAME="E50:39_1"/><span class="lab"><font color="Green" title="E62">A89</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c17">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></span>;<br/>
<a NAME="E51:39_1"/><span class="lab"><font color="Green" title="E63">A90</font></span>: 
<font color="Maroon" title="c17">T</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:39_1"><span class="lab"><font color="Green" title="E62">A89</font></span></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></span>;<br/>
<a NAME="E52:39_1"/><span class="lab"><font color="Green" title="E64">A91</font></span>: 
( <font color="Maroon" title="c9">a9</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="c9">a9</font>,<font color="Maroon" title="c11">c9</font> &amp; <font color="Maroon" title="c11">c9</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="c9">a9</font>,<font color="Maroon" title="c11">c9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:39_1"><span class="lab"><font color="Green" title="E46">A34</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E53:39_1"/>
not <font color="Maroon" title="c4">C</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c17">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1_11"/>
<font color="Maroon" title="c4">C</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c17">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:39_1_11"/><span class="kw">then </span>
<font color="Maroon" title="c4">C</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="c9">a9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:39_1"><span class="lab"><font color="Green" title="E62">A89</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<a NAME="E3:39_1_11"/><span class="kw">then </span>
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:39_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="txt" href="#E52:39_1"><span class="lab"><font color="Green" title="E64">A91</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:39_1_11"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E15:39_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E17:39_1"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="txt" href="#E45:39_1"><span class="lab"><font color="Green" title="E59">A84</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E55:39_1"/><span class="lab"><font color="Green" title="E65">A92</font></span>: 
<font color="Maroon" title="c18">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E56:39_1"/><span class="lab"><font color="Green" title="E66">A93</font></span>: 
<font color="Maroon" title="c18">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:39_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E51:39_1"><span class="lab"><font color="Green" title="E63">A90</font></span></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></span>;<br/>
<a NAME="E57:39_1"/><span class="lab"><font color="Green" title="E67">A94</font></span>: 
<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c5">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:39_1"><span class="lab"><font color="Green" title="E52">A77</font></span></a>, <a class="txt" href="#E37:39_1"><span class="lab"><font color="Green" title="E53">A78</font></span></a>, <a class="txt" href="#E38:39_1"><span class="lab"><font color="Green" title="E54">A79</font></span></a>, <a class="txt" href="#E44:39_1"><span class="lab"><font color="Green" title="E58">A83</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/>
<a NAME="E58:39_1"/><span class="lab"><font color="Green" title="E68">A95</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1_12"/><span class="lab"><font color="Green" title="E69">A96</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:39_1_12"/>
<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c9">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:39_1"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E3:39_1_12"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E7:39_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E26:39_1"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E1:39_1_12"><span class="lab"><font color="Green" title="E69">A96</font></span></a>, <a class="ref" href="aff_1.html#T62" title="AFF_1:th.62">AFF_1:62</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:39_1_12"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E5:39_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E9:39_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E14:39_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="txt" href="#E16:39_1"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E59:39_1"/><span class="lab"><font color="Green" title="E69">A97</font></span>: 
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_13"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1_13"/>
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:39_1_13"/><span class="kw">then </span>
<font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E10:39_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E14:39_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="txt" href="#E15:39_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E58:39_1"><span class="lab"><font color="Green" title="E68">A95</font></span></a>, <a class="txt" href="#E55:39_1"><span class="lab"><font color="Green" title="E65">A92</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:39_1_13"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E10:39_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E11:39_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E12:39_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="txt" href="#E14:39_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="txt" href="#E20:39_1"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="ref" href="aff_1.html#T65" title="AFF_1:th.65">AFF_1:65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c19">R</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c16">p</font>;<br/>
<a NAME="E60:39_1"/><span class="lab"><font color="Green" title="E70">A98</font></span>: 
<font color="Maroon" title="c16">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_14"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1_14"/>
<font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:39_1_14"/><span class="kw">then </span>
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E25:39_1"><span class="lab"><font color="Green" title="E43">A27</font></span></a>, <a class="txt" href="#E27:39_1"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="txt" href="#E40:39_1"><span class="lab"><font color="Green" title="E56">A81</font></span></a>, <a class="txt" href="#E32:39_1"><span class="lab"><font color="Green" title="E50">A75</font></span></a>, <a class="txt" href="#E43:39_1"><span class="lab"><font color="Green" title="E57">A82</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:39_1_14"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E10:39_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E14:39_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="txt" href="#E16:39_1"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="txt" href="#E58:39_1"><span class="lab"><font color="Green" title="E68">A95</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E61:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A99</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c16">p</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E62:39_1"/><span class="lab"><font color="Green" title="E72">A100</font></span>: 
<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c18">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:39_1"><span class="lab"><font color="Green" title="E64">A91</font></span></a>, <a class="txt" href="#E49:39_1"><span class="lab"><font color="Green" title="E61">A88</font></span></a>, <a class="txt" href="#E50:39_1"><span class="lab"><font color="Green" title="E62">A89</font></span></a>, <a class="txt" href="#E56:39_1"><span class="lab"><font color="Green" title="E66">A93</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/>
<a NAME="E63:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A101</font></span>: 
<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c18">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c5">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E12:39_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="txt" href="#E15:39_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E27:39_1"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="txt" href="#E40:39_1"><span class="lab"><font color="Green" title="E56">A81</font></span></a>, <a class="txt" href="#E32:39_1"><span class="lab"><font color="Green" title="E50">A75</font></span></a>, <a class="txt" href="#E43:39_1"><span class="lab"><font color="Green" title="E57">A82</font></span></a>, <a class="txt" href="#E55:39_1"><span class="lab"><font color="Green" title="E65">A92</font></span></a>, <a class="txt" href="#E57:39_1"><span class="lab"><font color="Green" title="E67">A94</font></span></a>, <a class="ref" href="aff_2.html#D1" target="_self" title="AFF_2:def.1">Def1</a></span>;<br/>
<a NAME="E64:39_1"/><span class="lab"><font color="Green" title="E74">A102</font></span>: 
( <font color="Maroon" title="c6">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="c6">a</font>,<font color="Maroon" title="c16">p</font> &amp; <font color="Maroon" title="c16">p</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="c6">a</font>,<font color="Maroon" title="c16">p</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:39_1"><span class="lab"><font color="Green" title="E70">A98</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E65:39_1"/>
not <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E20:39_1"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="txt" href="#E31:39_1"><span class="lab"><font color="Green" title="E49">A40</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/>
<a NAME="E66:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A103</font></span>: 
<font color="Maroon" title="c15">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:39_1"><span class="lab"><font color="Green" title="E52">A77</font></span></a>, <a class="txt" href="#E38:39_1"><span class="lab"><font color="Green" title="E54">A79</font></span></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></span>;<br/>
<a NAME="E67:39_1"/>
not <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c18">q</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c16">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_15"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1_15"/>
<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c18">q</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c16">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:39_1_15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A104</font></span>: 
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c5">o</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c16">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:39_1"><span class="lab"><font color="Green" title="E73">A101</font></span></a>, <a class="txt" href="#E59:39_1"><span class="lab"><font color="Green" title="E69">A97</font></span></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a></span>;<br/>
<a NAME="E3:39_1_15"/>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c5">o</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a>, <a class="ref" href="aff_1.html#T55" title="AFF_1:th.55">AFF_1:55</a></span>;<br/>
<a NAME="E4:39_1_15"/><span class="kw">then </span>
<font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E45:39_1"><span class="lab"><font color="Green" title="E59">A84</font></span></a>, <a class="txt" href="#E64:39_1"><span class="lab"><font color="Green" title="E74">A102</font></span></a>, <a class="txt" href="#E2:39_1_15"><span class="lab"><font color="Green" title="E76">A104</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a>, <a class="ref" href="aff_1.html#T67" title="AFF_1:th.67">AFF_1:67</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:39_1_15"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E37:39_1"><span class="lab"><font color="Green" title="E53">A78</font></span></a>, <a class="txt" href="#E39:39_1"><span class="lab"><font color="Green" title="E55">A80</font></span></a>, <a class="txt" href="#E44:39_1"><span class="lab"><font color="Green" title="E58">A83</font></span></a>, <a class="txt" href="#E46:39_1"><span class="lab"><font color="Green" title="E60">A86</font></span></a>, <a class="txt" href="#E66:39_1"><span class="lab"><font color="Green" title="E75">A103</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E69:39_1"/><span class="lab"><font color="Green" title="E76">A105</font></span>: 
<font color="Maroon" title="c20">r</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="c6">a</font>,<font color="Maroon" title="c16">p</font>
 <span class="kw">and </span><br/><a NAME="E70:39_1"/><span class="lab"><font color="Green" title="E77">A106</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c20">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E61:39_1"><span class="lab"><font color="Green" title="E71">A99</font></span></a>, <a class="ref" href="aff_1.html#T73" title="AFF_1:th.73">AFF_1:73</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E78">A107</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_16"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:39_1_16"/>
<font color="Maroon" title="c20">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c10">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font></span><br/><a NAME="E2:39_1_16"/><span class="kw">then </span>
<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c20">r</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c5">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E12:39_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="txt" href="#E15:39_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E27:39_1"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="txt" href="#E40:39_1"><span class="lab"><font color="Green" title="E56">A81</font></span></a>, <a class="txt" href="#E32:39_1"><span class="lab"><font color="Green" title="E50">A75</font></span></a>, <a class="txt" href="#E43:39_1"><span class="lab"><font color="Green" title="E57">A82</font></span></a>, <a class="txt" href="#E55:39_1"><span class="lab"><font color="Green" title="E65">A92</font></span></a>, <a class="txt" href="#E57:39_1"><span class="lab"><font color="Green" title="E67">A94</font></span></a>, <a class="txt" href="#E62:39_1"><span class="lab"><font color="Green" title="E72">A100</font></span></a>, <a class="ref" href="aff_2.html#D1" target="_self" title="AFF_2:def.1">Def1</a></span>;<br/><a NAME="E3:39_1_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A108</font></span>: 
<font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c10">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><a NAME="E4:39_1_16"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E7:39_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><a NAME="E5:39_1_16"/><span class="kw">then </span>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:39_1_16"/>
<font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c10">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:39_1"><span class="lab"><font color="Green" title="E59">A84</font></span></a>, <a class="txt" href="#E3:39_1_16"><span class="lab"><font color="Green" title="E79">A108</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E72:39_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c16">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:39_1"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="txt" href="#E40:39_1"><span class="lab"><font color="Green" title="E56">A81</font></span></a>, <a class="txt" href="#E32:39_1"><span class="lab"><font color="Green" title="E50">A75</font></span></a>, <a class="txt" href="#E43:39_1"><span class="lab"><font color="Green" title="E57">A82</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/>
<a NAME="E73:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c9">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c16">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E74:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c10">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E75:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A109</font></span>: 
<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c10">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:39_1"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="txt" href="#E26:39_1"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E76:39_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c9">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E7:39_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/>
<a NAME="E77:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E78:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A110</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c5">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E79:39_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:39_1"><span class="lab"><font color="Green" title="E77">A106</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E80:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c20">r</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E81:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c18">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c18">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E82:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c18">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c5">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:39_1"><span class="lab"><font color="Green" title="E73">A101</font></span></a>, <a class="txt" href="#E59:39_1"><span class="lab"><font color="Green" title="E69">A97</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E83:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A111</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c18">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:39_1"><span class="lab"><font color="Green" title="E59">A84</font></span></a>, <a class="txt" href="#E78:39_1"><span class="lab"><font color="Green" title="E80">A110</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E84:39_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c18">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:39_1"><span class="lab"><font color="Green" title="E77">A106</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E85:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c10">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c18">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E86:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E83:39_1"><span class="lab"><font color="Green" title="E81">A111</font></span></a>, <a class="txt" href="#E71:39_1"><span class="lab"><font color="Green" title="E78">A107</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E87:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A112</font></span>: 
<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E9:39_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E10:39_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E14:39_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="txt" href="#E61:39_1"><span class="lab"><font color="Green" title="E71">A99</font></span></a>, <a class="txt" href="#E64:39_1"><span class="lab"><font color="Green" title="E74">A102</font></span></a>, <a class="txt" href="#E69:39_1"><span class="lab"><font color="Green" title="E76">A105</font></span></a>, <a class="txt" href="#E75:39_1"><span class="lab"><font color="Green" title="E79">A109</font></span></a>, <a class="ref" href="aff_2.html#D1" target="_self" title="AFF_2:def.1">Def1</a></span>;<br/>
<a NAME="E88:39_1"/>
<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c18">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:39_1"><span class="lab"><font color="Green" title="E37">A20</font></span></a>, <a class="txt" href="#E28:39_1"><span class="lab"><font color="Green" title="E46">A34</font></span></a>, <a class="txt" href="#E62:39_1"><span class="lab"><font color="Green" title="E72">A100</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E89:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A113</font></span>: 
<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E11:39_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E15:39_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="#E55:39_1"><span class="lab"><font color="Green" title="E65">A92</font></span></a>, <a class="txt" href="#E61:39_1"><span class="lab"><font color="Green" title="E71">A99</font></span></a>, <a class="txt" href="#E64:39_1"><span class="lab"><font color="Green" title="E74">A102</font></span></a>, <a class="txt" href="#E69:39_1"><span class="lab"><font color="Green" title="E76">A105</font></span></a>, <a class="txt" href="#E83:39_1"><span class="lab"><font color="Green" title="E81">A111</font></span></a>, <a class="ref" href="aff_2.html#D1" target="_self" title="AFF_2:def.1">Def1</a></span>;<br/>
<a NAME="E90:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:39_1"><span class="lab"><font color="Green" title="E60">A86</font></span></a>, <a class="txt" href="#E87:39_1"><span class="lab"><font color="Green" title="E82">A112</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E91:39_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E92:39_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c20">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E93:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c20">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E94:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A114</font></span>: 
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E95:39_1"/>
<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:39_1"><span class="lab"><font color="Green" title="E60">A86</font></span></a>, <a class="txt" href="#E57:39_1"><span class="lab"><font color="Green" title="E67">A94</font></span></a>, <a class="txt" href="#E89:39_1"><span class="lab"><font color="Green" title="E83">A113</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E96:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:39_1"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="txt" href="#E94:39_1"><span class="lab"><font color="Green" title="E84">A114</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E97:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c5">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E87:39_1"><span class="lab"><font color="Green" title="E82">A112</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E98:39_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:39_1"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="txt" href="#E46:39_1"><span class="lab"><font color="Green" title="E60">A86</font></span></a>, <a class="txt" href="#E57:39_1"><span class="lab"><font color="Green" title="E67">A94</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
