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

<span class="kw">let </span><font color="Maroon" title="c1">AS</font> be   <a href="diraf.html#NM1" title="DIRAF:NM.1">AffinSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">q</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">AS</font><br/> for <font color="Olive" title="b8">A</font>, <font color="Olive" title="b9">P</font>, <font color="Olive" title="b10">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b8">A</font> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">P</font> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">C</font> &amp; not <font color="Olive" title="b8">A</font>,<font color="Olive" title="b9">P</font>,<font color="Olive" title="b10">C</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">c</font> &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b8">A</font> &amp; <font color="Olive" title="b5">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b8">A</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">P</font> &amp; <font color="Olive" title="b6">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">P</font> &amp; <font color="Olive" title="b4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">C</font> &amp; <font color="Olive" title="b7">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">C</font> &amp; <font color="Olive" title="b8">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b9">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b10">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b8">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b9">P</font> &amp; <font color="Olive" title="b8">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b10">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="c2">q</font>, <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">a9</font>, <font color="Maroon" title="c7">b9</font>, <font color="Maroon" title="c8">c9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">P</font>, <font color="Olive" title="b3">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">P</font> &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">C</font> &amp; not <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">P</font>,<font color="Olive" title="b3">C</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c6">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">P</font> &amp; <font color="Maroon" title="c7">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">P</font> &amp; <font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">C</font> &amp; <font color="Maroon" title="c8">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">C</font> &amp; <font color="Olive" title="b1">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b3">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">C</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c7">b9</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c8">c9</font> holds <br/><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">c9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c9">A</font>, <font color="Maroon" title="c10">P</font>, <font color="Maroon" title="c11">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font> &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">C</font> &amp; not <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">C</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c6">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font> &amp; <font color="Maroon" title="c7">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font> &amp; <font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">C</font> &amp; <font color="Maroon" title="c8">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">C</font> &amp; <font color="Maroon" title="c9">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c10">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c11">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c9">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">P</font> &amp; <font color="Maroon" title="c9">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">C</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c7">b9</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c8">c9</font> implies <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">c9</font> )</span><br/>





















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:62"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
<font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font>
 <span class="kw">and </span><br/><a NAME="E2:62"/><span class="lab"><font color="Green" title="E53">A2</font></span>: 
<font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font>
 <span class="kw">and </span><br/><a NAME="E3:62"/><span class="lab"><font color="Green" title="E54">A3</font></span>: 
<font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">C</font>
 <span class="kw">and </span><br/><a NAME="E4:62"/><span class="lab"><font color="Green" title="E55">A4</font></span>: 
not <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">C</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a> 
 <span class="kw">and </span><br/><a NAME="E5:62"/><span class="lab"><font color="Green" title="E56">A5</font></span>: 
<font color="Maroon" title="c2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">and </span><br/><a NAME="E6:62"/><span class="lab"><font color="Green" title="E57">A6</font></span>: 
<font color="Maroon" title="c2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">and </span><br/><a NAME="E7:62"/><span class="lab"><font color="Green" title="E58">A7</font></span>: 
<font color="Maroon" title="c2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font>
 <span class="kw">and </span><br/><a NAME="E8:62"/><span class="lab"><font color="Green" title="E59">A8</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font>
 <span class="kw">and </span><br/><a NAME="E9:62"/><span class="lab"><font color="Green" title="E60">A9</font></span>: 
<font color="Maroon" title="c6">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font>
 <span class="kw">and </span><br/><a NAME="E10:62"/><span class="lab"><font color="Green" title="E61">A10</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font>
 <span class="kw">and </span><br/><a NAME="E11:62"/><span class="lab"><font color="Green" title="E62">A11</font></span>: 
<font color="Maroon" title="c7">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font>
 <span class="kw">and </span><br/><a NAME="E12:62"/><span class="lab"><font color="Green" title="E63">A12</font></span>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">C</font>
 <span class="kw">and </span><br/><a NAME="E13:62"/><span class="lab"><font color="Green" title="E64">A13</font></span>: 
<font color="Maroon" title="c8">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">C</font>
 <span class="kw">and </span><br/><a NAME="E14:62"/><span class="lab"><font color="Green" title="E65">A14</font></span>: 
<font color="Maroon" title="c9">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="E15:62"/><span class="lab"><font color="Green" title="E66">A15</font></span>: 
<font color="Maroon" title="c10">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="E16:62"/><span class="lab"><font color="Green" title="E67">A16</font></span>: 
<font color="Maroon" title="c11">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="E17:62"/><span class="lab"><font color="Green" title="E68">A17</font></span>: 
<font color="Maroon" title="c9">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">P</font>
 <span class="kw">and </span><br/><a NAME="E18:62"/><span class="lab"><font color="Green" title="E69">A18</font></span>: 
<font color="Maroon" title="c9">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">C</font>
 <span class="kw">and </span><br/><a NAME="E19:62"/><span class="lab"><font color="Green" title="E70">A19</font></span>: 
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c7">b9</font>
 <span class="kw">and </span><br/><a NAME="E20:62"/><span class="lab"><font color="Green" title="E71">A20</font></span>: 
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c8">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="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">c9</font></span><br/>

<a NAME="E21:62"/><span class="lab"><font color="Green" title="E72">A21</font></span>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E5:62"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="txt" href="#E8:62"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E63">A12</font></span></a>, <a class="txt" href="#E14:62"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="txt" href="#E16:62"><span class="lab"><font color="Green" title="E67">A16</font></span></a>, <a class="txt" href="#E18:62"><span class="lab"><font color="Green" title="E69">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="E22:62"/><span class="lab"><font color="Green" title="E73">A22</font></span>: 
<font color="Maroon" title="c10">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E4:62"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="txt" href="#E14:62"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="txt" href="#E15:62"><span class="lab"><font color="Green" title="E66">A15</font></span></a>, <a class="ref" href="aff_4.html#T47" target="_self" title="AFF_4:th.47">Th47</a></span>;<br/>
<a NAME="E23:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A23</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E57">A6</font></span></a>, <a class="txt" href="#E10:62"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E63">A12</font></span></a>, <a class="txt" href="#E15:62"><span class="lab"><font color="Green" title="E66">A15</font></span></a>, <a class="txt" href="#E16:62"><span class="lab"><font color="Green" title="E67">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">consider </span><font color="Maroon" title="c12">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><span class="kw"> such that </span><br/><a NAME="E25:62"/><span class="lab"><font color="Green" title="E75">A24</font></span>: 
( <font color="Maroon" title="c10">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">X</font> &amp; <font color="Maroon" title="c11">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">X</font> )
 <span class="kw">and </span><br/><a NAME="E26:62"/><span class="lab"><font color="Green" title="E76">A25</font></span>: 
<font color="Maroon" title="c12">X</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E15:62"><span class="lab"><font color="Green" title="E66">A15</font></span></a>, <a class="txt" href="#E16:62"><span class="lab"><font color="Green" title="E67">A16</font></span></a>, <a class="ref" href="aff_4.html#T38" target="_self" title="AFF_4:th.38">Th38</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><span class="kw"> such that </span><br/><a NAME="E28:62"/><span class="lab"><font color="Green" title="E77">A26</font></span>: 
<font color="Maroon" title="c9">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">Y</font>
 <span class="kw">and </span><br/><a NAME="E29:62"/><span class="lab"><font color="Green" title="E78">A27</font></span>: 
<font color="Maroon" title="c11">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">Y</font>
 <span class="kw">and </span><br/><a NAME="E30:62"/><span class="lab"><font color="Green" title="E79">A28</font></span>: 
<font color="Maroon" title="c13">Y</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E14:62"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="txt" href="#E16:62"><span class="lab"><font color="Green" title="E67">A16</font></span></a>, <a class="ref" href="aff_4.html#T38" target="_self" title="AFF_4:th.38">Th38</a></span>;<br/>
<a NAME="E31:62"/><span class="lab"><font color="Green" title="E80">A29</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E5:62"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="txt" href="#E8:62"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="txt" href="#E10:62"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="txt" href="#E14:62"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="txt" href="#E15:62"><span class="lab"><font color="Green" title="E66">A15</font></span></a>, <a class="txt" href="#E17:62"><span class="lab"><font color="Green" title="E68">A17</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E81">A30</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:62_1"/><span class="lab"><font color="Green" title="E82">A31</font></span>: 
<font color="Maroon" title="c2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a9</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="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">c9</font></span><br/><a NAME="E2:62_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A32</font></span>: 
<font color="Maroon" title="c2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E5:62"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="txt" href="#E7:62"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="txt" href="#E8:62"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="txt" href="#E9:62"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E63">A12</font></span></a>, <a class="txt" href="#E14:62"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="txt" href="#E16:62"><span class="lab"><font color="Green" title="E67">A16</font></span></a>, <a class="txt" href="#E18:62"><span class="lab"><font color="Green" title="E69">A18</font></span></a>, <a class="txt" href="#E20:62"><span class="lab"><font color="Green" title="E71">A20</font></span></a>, <a class="ref" href="aff_4.html#T8" target="_self" title="AFF_4:th.8">Th8</a></span>;<br/><div><span class="lab"><font color="Green" title="E84">A33</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c14">BC</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>;<br/><span class="kw">set </span><font color="Maroon" title="c15">BC9</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">c9</font>;<br/><span class="kw">set </span><font color="Maroon" title="c16">AB</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>;<br/><span class="kw">set </span><font color="Maroon" title="c17">AB9</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c7">b9</font>;<br/><span class="kw">set </span><font color="Maroon" title="c18">AC</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>;<br/><span class="kw">set </span><font color="Maroon" title="c19">AC9</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c8">c9</font>;<br/><span class="kw">assume </span><a NAME="E1:62_1_1"/><span class="lab"><font color="Green" title="E85">A34</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a9</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="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">c9</font></span><br/><span class="kw">assume </span><a NAME="E2:62_1_1"/><span class="lab"><font color="Green" title="E86">A35</font></span>: 
not <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">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="E3:62_1_1"/><span class="lab"><font color="Green" title="E87">A36</font></span>: 
( <font color="Maroon" title="c7">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="c7">b9</font>,<font color="Maroon" title="c8">c9</font> &amp; <font color="Maroon" title="c8">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="c7">b9</font>,<font color="Maroon" title="c8">c9</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/><a NAME="E4:62_1_1"/><span class="lab"><font color="Green" title="E88">A37</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:62"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E63">A12</font></span></a>, <a class="txt" href="#E25:62"><span class="lab"><font color="Green" title="E75">A24</font></span></a>, <a class="txt" href="#E26:62"><span class="lab"><font color="Green" title="E76">A25</font></span></a>, <a class="txt" href="#E23:62"><span class="lab"><font color="Green" title="E74">A23</font></span></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></span>;<br/><a NAME="E5:62_1_1"/><span class="lab"><font color="Green" title="E89">A38</font></span>: 
<font color="Maroon" title="c5">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="c4">b</font>,<font color="Maroon" title="c5">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/><a NAME="E6:62_1_1"/><span class="lab"><font color="Green" title="E90">A39</font></span>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:62"><span class="lab"><font color="Green" title="E74">A23</font></span></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/><a NAME="E7:62_1_1"/><span class="lab"><font color="Green" title="E91">A40</font></span>: 
<font color="Maroon" title="c8">c9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E11:62"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="txt" href="#E13:62"><span class="lab"><font color="Green" title="E64">A13</font></span></a>, <a class="txt" href="#E15:62"><span class="lab"><font color="Green" title="E66">A15</font></span></a>, <a class="txt" href="#E16:62"><span class="lab"><font color="Green" title="E67">A16</font></span></a>, <a class="txt" href="#E22:62"><span class="lab"><font color="Green" title="E73">A22</font></span></a>, <a class="txt" href="#E2:62_1"><span class="lab"><font color="Green" title="E83">A32</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/><a NAME="E8:62_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A41</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">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="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/><a NAME="E9:62_1_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">c9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:62"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="txt" href="#E13:62"><span class="lab"><font color="Green" title="E64">A13</font></span></a>, <a class="txt" href="#E25:62"><span class="lab"><font color="Green" title="E75">A24</font></span></a>, <a class="txt" href="#E26:62"><span class="lab"><font color="Green" title="E76">A25</font></span></a>, <a class="txt" href="#E7:62_1_1"><span class="lab"><font color="Green" title="E91">A40</font></span></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><span class="kw"> such that </span><br/><a NAME="E11:62_1_1"/><span class="lab"><font color="Green" title="E93">A42</font></span>: 
<font color="Maroon" title="c20">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="c4">b</font>,<font color="Maroon" title="c5">c</font>
 <span class="kw">and </span><br/><a NAME="E12:62_1_1"/><span class="lab"><font color="Green" title="E94">A43</font></span>: 
<font color="Maroon" title="c20">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="c7">b9</font>,<font color="Maroon" title="c8">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:62"><span class="lab"><font color="Green" title="E76">A25</font></span></a>, <a class="txt" href="#E2:62_1_1"><span class="lab"><font color="Green" title="E86">A35</font></span></a>, <a class="txt" href="#E8:62_1_1"><span class="lab"><font color="Green" title="E92">A41</font></span></a>, <a class="txt" href="#E6:62_1_1"><span class="lab"><font color="Green" title="E90">A39</font></span></a>, <a class="txt" href="#E5:62_1_1"><span class="lab"><font color="Green" title="E89">A38</font></span></a>, <a class="txt" href="#E3:62_1_1"><span class="lab"><font color="Green" title="E87">A36</font></span></a>, <a class="txt" href="#E4:62_1_1"><span class="lab"><font color="Green" title="E88">A37</font></span></a>, <a class="ref" href="aff_4.html#T22" target="_self" title="AFF_4:th.22">Th22</a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/><a NAME="E13:62_1_1"/><span class="lab"><font color="Green" title="E95">A44</font></span>: 
<font color="Maroon" title="c6">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="c6">a9</font>,<font color="Maroon" title="c8">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/><a NAME="E14:62_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">c9</font>,<font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c20">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:62_1_1"><span class="lab"><font color="Green" title="E92">A41</font></span></a>, <a class="txt" href="#E3:62_1_1"><span class="lab"><font color="Green" title="E87">A36</font></span></a>, <a class="txt" href="#E12:62_1_1"><span class="lab"><font color="Green" title="E94">A43</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><span class="kw"> such that </span><br/><a NAME="E16:62_1_1"/><span class="lab"><font color="Green" title="E96">A45</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">c9</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c21">y</font>
 <span class="kw">and </span><br/><a NAME="E17:62_1_1"/><span class="lab"><font color="Green" title="E97">A46</font></span>: 
<font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c6">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">p</font>,<font color="Maroon" title="c21">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:62_1_1"><span class="lab"><font color="Green" title="E91">A40</font></span></a>, <a class="ref" href="aff_4.html#T1" target="_self" title="AFF_4:th.1">Th1</a></span>;<br/><a NAME="E18:62_1_1"/><span class="lab"><font color="Green" title="E98">A47</font></span>: 
<font color="Maroon" title="c5">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="c3">a</font>,<font color="Maroon" title="c5">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/><a NAME="E19:62_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c20">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62_1_1"><span class="lab"><font color="Green" title="E90">A39</font></span></a>, <a class="txt" href="#E5:62_1_1"><span class="lab"><font color="Green" title="E89">A38</font></span></a>, <a class="txt" href="#E11:62_1_1"><span class="lab"><font color="Green" title="E93">A42</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><span class="kw"> such that </span><br/><a NAME="E21:62_1_1"/><span class="lab"><font color="Green" title="E99">A48</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c22">x</font>
 <span class="kw">and </span><br/><a NAME="E22:62_1_1"/><span class="lab"><font color="Green" title="E100">A49</font></span>: 
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">p</font>,<font color="Maroon" title="c22">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:62"><span class="lab"><font color="Green" title="E74">A23</font></span></a>, <a class="ref" href="aff_4.html#T1" target="_self" title="AFF_4:th.1">Th1</a></span>;<br/><a NAME="E23:62_1_1"/><span class="lab"><font color="Green" title="E101">A50</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/><a NAME="E24:62_1_1"/><span class="lab"><font color="Green" title="E102">A51</font></span>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:62"><span class="lab"><font color="Green" title="E72">A21</font></span></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/><a NAME="E25:62_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A52</font></span>: 
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:62"><span class="lab"><font color="Green" title="E72">A21</font></span></a>, <a class="txt" href="#E18:62_1_1"><span class="lab"><font color="Green" title="E98">A47</font></span></a>, <a class="txt" href="#E21:62_1_1"><span class="lab"><font color="Green" title="E99">A48</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">set </span><font color="Maroon" title="c23">K</font> = <font color="Maroon" title="c20">p</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span>;<br/><a NAME="E26:62_1_1"/><span class="lab"><font color="Green" title="E104">A53</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/><a NAME="E27:62_1_1"/><span class="lab"><font color="Green" title="E105">A54</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">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="#E31:62"><span class="lab"><font color="Green" title="E80">A29</font></span></a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/><a NAME="E28:62_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A55</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c20">p</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</a></span>;<br/><a NAME="E29:62_1_1"/><span class="lab"><font color="Green" title="E107">A56</font></span>: 
<font color="Maroon" title="c20">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">p</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:62_1_1"><span class="lab"><font color="Green" title="E105">A54</font></span></a>, <a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</a></span>;<br/><a NAME="E30:62_1_1"/>
<font color="Maroon" title="c20">p</font>,<font color="Maroon" title="c22">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:62_1_1"><span class="lab"><font color="Green" title="E100">A49</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><a NAME="E31:62_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">p</font>,<font color="Maroon" title="c22">x</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="c3">a</font>,<font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:62"><span class="lab"><font color="Green" title="E80">A29</font></span></a>, <a class="ref" href="aff_1.html#D4" title="AFF_1:def.4">AFF_1:def 4</a></span>;<br/><a NAME="E32:62_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">p</font>,<font color="Maroon" title="c22">x</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c20">p</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:62_1_1"><span class="lab"><font color="Green" title="E106">A55</font></span></a>, <a class="ref" href="aff_4.html#T3" target="_self" title="AFF_4:th.3">Th3</a></span>;<br/><a NAME="E33:62_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A57</font></span>: 
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">p</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:62_1_1"><span class="lab"><font color="Green" title="E107">A56</font></span></a>, <a class="ref" href="aff_4.html#T2" target="_self" title="AFF_4:th.2">Th2</a></span>;<br/><a NAME="E34:62_1_1"/><span class="lab"><font color="Green" title="E109">A58</font></span>: 
<font color="Maroon" title="c6">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E9:62"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="txt" href="#E11:62"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="txt" href="#E14:62"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="txt" href="#E15:62"><span class="lab"><font color="Green" title="E66">A15</font></span></a>, <a class="txt" href="#E17:62"><span class="lab"><font color="Green" title="E68">A17</font></span></a>, <a class="txt" href="#E1:62_1"><span class="lab"><font color="Green" title="E82">A31</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/><a NAME="E35:62_1_1"/>
<font color="Maroon" title="c20">p</font>,<font color="Maroon" title="c21">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c7">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:62_1_1"><span class="lab"><font color="Green" title="E97">A46</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><a NAME="E36:62_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A59</font></span>: 
<font color="Maroon" title="c20">p</font>,<font color="Maroon" title="c21">y</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">a9</font>,<font color="Maroon" title="c7">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:62_1_1"><span class="lab"><font color="Green" title="E109">A58</font></span></a>, <a class="ref" href="aff_1.html#D4" title="AFF_1:def.4">AFF_1:def 4</a></span>;<br/><a NAME="E37:62_1_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">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="c6">a9</font>,<font color="Maroon" title="c7">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:62"><span class="lab"><font color="Green" title="E70">A19</font></span></a>, <a class="txt" href="#E31:62"><span class="lab"><font color="Green" title="E80">A29</font></span></a>, <a class="txt" href="#E34:62_1_1"><span class="lab"><font color="Green" title="E109">A58</font></span></a>, <a class="ref" href="aff_1.html#T51" title="AFF_1:th.51">AFF_1:51</a></span>;<br/><a NAME="E38:62_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c7">b9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c20">p</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:62_1_1"><span class="lab"><font color="Green" title="E106">A55</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/><a NAME="E39:62_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">p</font>,<font color="Maroon" title="c21">y</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c20">p</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:62_1_1"><span class="lab"><font color="Green" title="E110">A59</font></span></a>, <a class="ref" href="aff_4.html#T3" target="_self" title="AFF_4:th.3">Th3</a></span>;<br/><a NAME="E40:62_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A60</font></span>: 
<font color="Maroon" title="c21">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">p</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:62_1_1"><span class="lab"><font color="Green" title="E107">A56</font></span></a>, <a class="ref" href="aff_4.html#T2" target="_self" title="AFF_4:th.2">Th2</a></span>;<br/><a NAME="E41:62_1_1"/><span class="lab"><font color="Green" title="E112">A61</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:62"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E63">A12</font></span></a>, <a class="txt" href="#E28:62"><span class="lab"><font color="Green" title="E77">A26</font></span></a>, <a class="txt" href="#E29:62"><span class="lab"><font color="Green" title="E78">A27</font></span></a>, <a class="txt" href="#E30:62"><span class="lab"><font color="Green" title="E79">A28</font></span></a>, <a class="txt" href="#E21:62"><span class="lab"><font color="Green" title="E72">A21</font></span></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></span>;<br/><a NAME="E42:62_1_1"/><span class="lab"><font color="Green" title="E113">A62</font></span>: 
<font color="Maroon" title="c8">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="c6">a9</font>,<font color="Maroon" title="c8">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/><a NAME="E43:62_1_1"/><span class="lab"><font color="Green" title="E114">A63</font></span>: 
<font color="Maroon" title="c6">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E9:62"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="txt" href="#E13:62"><span class="lab"><font color="Green" title="E64">A13</font></span></a>, <a class="txt" href="#E14:62"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="txt" href="#E16:62"><span class="lab"><font color="Green" title="E67">A16</font></span></a>, <a class="txt" href="#E18:62"><span class="lab"><font color="Green" title="E69">A18</font></span></a>, <a class="txt" href="#E1:62_1"><span class="lab"><font color="Green" title="E82">A31</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/><a NAME="E44:62_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c8">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="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/><a NAME="E45:62_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A64</font></span>: 
<font color="Maroon" title="c21">y</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">a9</font>,<font color="Maroon" title="c8">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:62_1_1"><span class="lab"><font color="Green" title="E114">A63</font></span></a>, <a class="txt" href="#E13:62_1_1"><span class="lab"><font color="Green" title="E95">A44</font></span></a>, <a class="txt" href="#E42:62_1_1"><span class="lab"><font color="Green" title="E113">A62</font></span></a>, <a class="txt" href="#E16:62_1_1"><span class="lab"><font color="Green" title="E96">A45</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/><a NAME="E46:62_1_1"/><span class="lab"><font color="Green" title="E116">A65</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c8">c9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:62"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="txt" href="#E13:62"><span class="lab"><font color="Green" title="E64">A13</font></span></a>, <a class="txt" href="#E28:62"><span class="lab"><font color="Green" title="E77">A26</font></span></a>, <a class="txt" href="#E29:62"><span class="lab"><font color="Green" title="E78">A27</font></span></a>, <a class="txt" href="#E30:62"><span class="lab"><font color="Green" title="E79">A28</font></span></a>, <a class="txt" href="#E43:62_1_1"><span class="lab"><font color="Green" title="E114">A63</font></span></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></span>;<br/><div><span class="lab"><font color="Green" title="E117">A66</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:62_1_1_1"/><span class="lab"><font color="Green" title="E118">A67</font></span>: 
<font color="Maroon" title="c22">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">y</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="E2:62_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">p</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c22">x</font>,<font color="Maroon" title="c21">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:62_1_1"><span class="lab"><font color="Green" title="E105">A54</font></span></a>, <a class="txt" href="#E33:62_1_1"><span class="lab"><font color="Green" title="E108">A57</font></span></a>, <a class="txt" href="#E40:62_1_1"><span class="lab"><font color="Green" title="E111">A60</font></span></a>, <a class="ref" href="aff_4.html#T27" target="_self" title="AFF_4:th.27">Th27</a>, <a class="ref" href="aff_1.html#T71" title="AFF_1:th.71">AFF_1:71</a></span>;<br/><a NAME="E3:62_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">p</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:62"><span class="lab"><font color="Green" title="E79">A28</font></span></a>, <a class="txt" href="#E41:62_1_1"><span class="lab"><font color="Green" title="E112">A61</font></span></a>, <a class="txt" href="#E46:62_1_1"><span class="lab"><font color="Green" title="E116">A65</font></span></a>, <a class="txt" href="#E25:62_1_1"><span class="lab"><font color="Green" title="E103">A52</font></span></a>, <a class="txt" href="#E45:62_1_1"><span class="lab"><font color="Green" title="E115">A64</font></span></a>, <a class="txt" href="#E1:62_1_1_1"><span class="lab"><font color="Green" title="E118">A67</font></span></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></span>;<br/><a NAME="E4:62_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A68</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:62"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="txt" href="#E28:62"><span class="lab"><font color="Green" title="E77">A26</font></span></a>, <a class="txt" href="#E30:62"><span class="lab"><font color="Green" title="E79">A28</font></span></a>, <a class="txt" href="#E23:62_1_1"><span class="lab"><font color="Green" title="E101">A50</font></span></a>, <a class="txt" href="#E28:62_1_1"><span class="lab"><font color="Green" title="E106">A55</font></span></a>, <a class="ref" href="aff_4.html#T23" target="_self" title="AFF_4:th.23">Th23</a></span>;<br/><a NAME="E5:62_1_1_1"/>
<font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E57">A6</font></span></a>, <a class="txt" href="#E10:62"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="txt" href="#E15:62"><span class="lab"><font color="Green" title="E66">A15</font></span></a>, <a class="ref" href="aff_1.html#T71" title="AFF_1:th.71">AFF_1:71</a></span>;<br/><a NAME="E6:62_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E57">A6</font></span></a>, <a class="txt" href="#E28:62"><span class="lab"><font color="Green" title="E77">A26</font></span></a>, <a class="txt" href="#E30:62"><span class="lab"><font color="Green" title="E79">A28</font></span></a>, <a class="txt" href="#E26:62_1_1"><span class="lab"><font color="Green" title="E104">A53</font></span></a>, <a class="txt" href="#E4:62_1_1_1"><span class="lab"><font color="Green" title="E119">A68</font></span></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></span>;<br/><span class="kw">hence </span><a NAME="E7:62_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="txt" href="#E28:62"><span class="lab"><font color="Green" title="E77">A26</font></span></a>, <a class="txt" href="#E29:62"><span class="lab"><font color="Green" title="E78">A27</font></span></a>, <a class="txt" href="#E30:62"><span class="lab"><font color="Green" title="E79">A28</font></span></a>, <a class="ref" href="aff_4.html#D5" target="_self" title="AFF_4:def.5">Def5</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="E48:62_1_1"/><span class="lab"><font color="Green" title="E118">A69</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">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">a9</font>,<font color="Maroon" title="c8">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:62"><span class="lab"><font color="Green" title="E71">A20</font></span></a>, <a class="txt" href="#E21:62"><span class="lab"><font color="Green" title="E72">A21</font></span></a>, <a class="txt" href="#E43:62_1_1"><span class="lab"><font color="Green" title="E114">A63</font></span></a>, <a class="ref" href="aff_1.html#T51" title="AFF_1:th.51">AFF_1:51</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:62_1_1_2"/>
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">y</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="E2:62_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">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="c3">a</font>,<font color="Maroon" title="c5">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:62_1_1"><span class="lab"><font color="Green" title="E95">A44</font></span></a>, <a class="txt" href="#E48:62_1_1"><span class="lab"><font color="Green" title="E118">A69</font></span></a>, <a class="txt" href="#E25:62_1_1"><span class="lab"><font color="Green" title="E103">A52</font></span></a>, <a class="txt" href="#E45:62_1_1"><span class="lab"><font color="Green" title="E115">A64</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/><a NAME="E3:62_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:62"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="txt" href="#E9:62"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="txt" href="#E14:62"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="txt" href="#E1:62_1_1"><span class="lab"><font color="Green" title="E85">A34</font></span></a>, <a class="txt" href="#E24:62_1_1"><span class="lab"><font color="Green" title="E102">A51</font></span></a>, <a class="txt" href="#E18:62_1_1"><span class="lab"><font color="Green" title="E98">A47</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="E4:62_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E7:62"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E63">A12</font></span></a>, <a class="txt" href="#E14:62"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="txt" href="#E16:62"><span class="lab"><font color="Green" title="E67">A16</font></span></a>, <a class="txt" href="#E18:62"><span class="lab"><font color="Green" title="E69">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><span class="kw">hence </span><a NAME="E50:62_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:62_1_1"><span class="lab"><font color="Green" title="E117">A66</font></span></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><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:62_1_2"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a9</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="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">c9</font></span><br/><a NAME="E2:62_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b9</font> &amp; <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E5:62"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E57">A6</font></span></a>, <a class="txt" href="#E7:62"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="txt" href="#E8:62"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="txt" href="#E10:62"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="txt" href="#E11:62"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E63">A12</font></span></a>, <a class="txt" href="#E13:62"><span class="lab"><font color="Green" title="E64">A13</font></span></a>, <a class="txt" href="#E14:62"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="txt" href="#E15:62"><span class="lab"><font color="Green" title="E66">A15</font></span></a>, <a class="txt" href="#E16:62"><span class="lab"><font color="Green" title="E67">A16</font></span></a>, <a class="txt" href="#E17:62"><span class="lab"><font color="Green" title="E68">A17</font></span></a>, <a class="txt" href="#E18:62"><span class="lab"><font color="Green" title="E69">A18</font></span></a>, <a class="txt" href="#E19:62"><span class="lab"><font color="Green" title="E70">A19</font></span></a>, <a class="txt" href="#E20:62"><span class="lab"><font color="Green" title="E71">A20</font></span></a>, <a class="ref" href="aff_4.html#T9" target="_self" title="AFF_4:th.9">Th9</a></span>;<br/><span class="kw">hence </span><a NAME="E3:62_1_2"/>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T11" title="AFF_1:th.11">AFF_1:11</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">hence </span><a NAME="E5:62_1"/>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1"><span class="lab"><font color="Green" title="E84">A33</font></span></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><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:62_2"/>
<font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a9</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="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">c9</font></span><br/><a NAME="E2:62_2"/><span class="kw">then </span>
( <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b9</font> &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E5:62"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E57">A6</font></span></a>, <a class="txt" href="#E7:62"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="txt" href="#E8:62"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="txt" href="#E10:62"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="txt" href="#E11:62"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E63">A12</font></span></a>, <a class="txt" href="#E13:62"><span class="lab"><font color="Green" title="E64">A13</font></span></a>, <a class="txt" href="#E14:62"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="txt" href="#E15:62"><span class="lab"><font color="Green" title="E66">A15</font></span></a>, <a class="txt" href="#E16:62"><span class="lab"><font color="Green" title="E67">A16</font></span></a>, <a class="txt" href="#E17:62"><span class="lab"><font color="Green" title="E68">A17</font></span></a>, <a class="txt" href="#E18:62"><span class="lab"><font color="Green" title="E69">A18</font></span></a>, <a class="txt" href="#E19:62"><span class="lab"><font color="Green" title="E70">A19</font></span></a>, <a class="txt" href="#E20:62"><span class="lab"><font color="Green" title="E71">A20</font></span></a>, <a class="ref" href="aff_4.html#T8" target="_self" title="AFF_4:th.8">Th8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:62_2"/>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">c9</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>; <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">hence </span><a NAME="E34:62"/>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:62"><span class="lab"><font color="Green" title="E81">A30</font></span></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>
