<?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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a>  iff <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V12" title="AFF_2:attr.12">satisfying_des_1</a>  )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V12" title="AFF_2:attr.12">satisfying_des_1</a>  implies <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:21_1"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V12" title="AFF_2:attr.12">satisfying_des_1</a> </span><br/><span class="kw">thus </span><a NAME="E2:21_1"/>
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V12" title="AFF_2:attr.12">satisfying_des_1</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_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="firebrick">:: according to </font></span><a class="ref" href="aff_2.html#D12" target="_self" title="AFF_2:def.12">AFF_2:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">a</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">a9</font>, <font color="Olive" title="b7">b9</font>, <font color="Olive" title="b8">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="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b3">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="b6">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="b4">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="b7">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="b5">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="b8">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="b3">a</font>,<font color="Olive" title="b4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">a9</font>,<font color="Olive" title="b7">b9</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">a9</font>,<font color="Olive" title="b8">c9</font> &amp; <font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">b9</font>,<font color="Olive" title="b8">c9</font> &amp;  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font> &amp; <font color="Olive" title="b5">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b8">c9</font> holds <br/><font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b2">C</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">a9</font>, <font color="Olive" title="b5">b9</font>, <font color="Olive" title="b6">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="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b1">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="b4">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="b2">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="b5">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="b3">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="b6">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="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">a9</font>,<font color="Olive" title="b5">b9</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">a9</font>,<font color="Olive" title="b6">c9</font> &amp; <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">b9</font>,<font color="Olive" title="b6">c9</font> &amp;  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">c9</font> holds <br/><font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">a9</font>, <font color="Maroon" title="c9">b9</font>, <font color="Maroon" title="c10">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c5">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="c8">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="c6">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="c9">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="c7">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="c10">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="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">c9</font> &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font> &amp;  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c9</font> implies <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font> )</span><br/>

















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:21_1_1"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E2:21_1_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c5">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="E3:21_1_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<font color="Maroon" title="c8">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="E4:21_1_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c6">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="E5:21_1_1"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<font color="Maroon" title="c9">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="E6:21_1_1"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
( <font color="Maroon" title="c7">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="c10">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="E7:21_1_1"/><span class="lab"><font color="Green" title="E23">A8</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="E8:21_1_1"/><span class="lab"><font color="Green" title="E24">A9</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="E9:21_1_1"/><span class="lab"><font color="Green" title="E25">A10</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="E10:21_1_1"/><span class="lab"><font color="Green" title="E26">A11</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="E11:21_1_1"/><span class="lab"><font color="Green" title="E27">A12</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="E12:21_1_1"/><span class="lab"><font color="Green" title="E28">A13</font></span>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font>
 <span class="kw">and </span><br/><a NAME="E13:21_1_1"/><span class="lab"><font color="Green" title="E29">A14</font></span>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">c9</font>
 <span class="kw">and </span><br/><a NAME="E14:21_1_1"/><span class="lab"><font color="Green" title="E30">A15</font></span>: 
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font>
 <span class="kw">and </span><br/><a NAME="E15:21_1_1"/><span class="lab"><font color="Green" title="E31">A16</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>
 <span class="kw">and </span><br/><a NAME="E16:21_1_1"/><span class="lab"><font color="Green" title="E32">A17</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font></span><br/>

<span class="kw">assume </span><a NAME="E17:21_1_1"/><span class="lab"><font color="Green" title="E33">A18</font></span>: 
 not <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c11">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="E19:21_1_1"/><span class="lab"><font color="Green" title="E34">A19</font></span>: 
<font color="Maroon" title="c10">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">K</font>
 <span class="kw">and </span><br/><a NAME="E20:21_1_1"/><span class="lab"><font color="Green" title="E35">A20</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c11">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="aff_1.html#T49" title="AFF_1:th.49">AFF_1:49</a></span>;<br/>
<a NAME="E21:21_1_1"/><span class="lab"><font color="Green" title="E36">A21</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21_1_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<a NAME="E22:21_1_1"/><span class="lab"><font color="Green" title="E37">A22</font></span>: 
 not <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c11">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_1_1_1"/>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c11">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</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="#E20:21_1_1"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="ref" href="aff_1.html#T43" title="AFF_1:th.43">AFF_1:43</a></span>;<br/>
<a NAME="E3:21_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A23</font></span>: 
<font color="Maroon" title="c7">c</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="#E2:21_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E7:21_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="aff_1.html#T23" title="AFF_1:th.23">AFF_1:23</a></span>;<br/>
<a NAME="E4:21_1_1_1"/>
<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21_1_1"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E5:21_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">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="#E2:21_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E7:21_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E21:21_1_1"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="txt" href="#E3:21_1_1_1"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="ref" href="aff_1.html#T27" title="AFF_1:th.27">AFF_1:27</a></span>;<br/>
<a NAME="E6:21_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">c9</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="#E3:21_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E7:21_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="aff_1.html#T23" title="AFF_1:th.23">AFF_1:23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:21_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E7:21_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E9:21_1_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E11:21_1_1"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E16:21_1_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E3:21_1_1_1"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E23:21_1_1"/><span class="lab"><font color="Green" title="E38">A24</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="c11">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_1_1_2"/><span class="lab"><font color="Green" title="E39">A25</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_1_1_2"/>
<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21_1_1"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E3:21_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E19:21_1_1"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="txt" href="#E20:21_1_1"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E22:21_1_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="txt" href="#E1:21_1_1_2"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="ref" href="aff_1.html#T32" title="AFF_1:th.32">AFF_1:32</a>, <a class="ref" href="aff_1.html#T40" title="AFF_1:th.40">AFF_1:40</a></span>;<br/>
<a NAME="E4:21_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:21_1_1"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E5:21_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c8">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b9</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:21_1_1"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E6:21_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c9">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E7:21_1_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">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E5:21_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E10:21_1_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a>, <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="E8:21_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21_1_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E39">A26</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:21_1_1_3"/>
<font color="Maroon" title="c9">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:21_1_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> or <font color="Maroon" title="c8">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:21_1_1"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E13:21_1_1"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:21_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E3:21_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E5:21_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E10:21_1_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E15:21_1_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E25:21_1_1"/><span class="lab"><font color="Green" title="E40">A27</font></span>: 
<font color="Maroon" title="c11">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="#E20:21_1_1"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="ref" href="aff_1.html#T36" title="AFF_1:th.36">AFF_1:36</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x</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="E27:21_1_1"/><span class="lab"><font color="Green" title="E41">A28</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">K</font>
 <span class="kw">and </span><br/><a NAME="E28:21_1_1"/><span class="lab"><font color="Green" title="E42">A29</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:21_1_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E29:21_1_1"/>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c12">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:21_1_1"><span class="lab"><font color="Green" title="E42">A29</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="E30:21_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c12">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21_1_1"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="txt" href="#E21:21_1_1"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E31:21_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c12">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E2:21_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E3:21_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E4:21_1_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E5:21_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E7:21_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E8:21_1_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E10:21_1_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E12:21_1_1"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E19:21_1_1"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="txt" href="#E20:21_1_1"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E25:21_1_1"><span class="lab"><font color="Green" title="E40">A27</font></span></a>, <a class="txt" href="#E27:21_1_1"><span class="lab"><font color="Green" title="E41">A28</font></span></a>, <a class="txt" href="#E23:21_1_1"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="ref" href="aff_2.html#D11" target="_self" title="AFF_2:def.11">Def11</a></span>;<br/>
<a NAME="E32:21_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c12">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> or <font color="Maroon" title="c9">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:21_1_1"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E33:21_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:21_1_1"><span class="lab"><font color="Green" title="E39">A26</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="E34:21_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A30</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E35:21_1_1"/><span class="lab"><font color="Green" title="E44">A31</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<a NAME="E36:21_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:21_1_1"><span class="lab"><font color="Green" title="E42">A29</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E37:21_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21_1_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E27:21_1_1"><span class="lab"><font color="Green" title="E41">A28</font></span></a>, <a class="txt" href="#E34:21_1_1"><span class="lab"><font color="Green" title="E43">A30</font></span></a>, <a class="txt" href="#E35:21_1_1"><span class="lab"><font color="Green" title="E44">A31</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E38:21_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E9:21_1_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E16:21_1_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E17:21_1_1"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E19:21_1_1"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="txt" href="#E20:21_1_1"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E25:21_1_1"><span class="lab"><font color="Green" title="E40">A27</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">assume </span><a NAME="E2:21"/><span class="lab"><font color="Green" title="E16">A32</font></span>: 
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V12" title="AFF_2:attr.12">satisfying_des_1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a> </span><br/>

<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="firebrick">:: according to </font></span><a class="ref" href="aff_2.html#D11" target="_self" title="AFF_2:def.11">AFF_2:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">a</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">a9</font>, <font color="Olive" title="b7">b9</font>, <font color="Olive" title="b8">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="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b3">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="b6">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="b4">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="b7">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="b5">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="b8">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="b3">a</font>,<font color="Olive" title="b4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">a9</font>,<font color="Olive" title="b7">b9</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">a9</font>,<font color="Olive" title="b8">c9</font> holds <br/><font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">b9</font>,<font color="Olive" title="b8">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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">a9</font>, <font color="Olive" title="b5">b9</font>, <font color="Olive" title="b6">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="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b1">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="b4">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="b2">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="b5">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="b3">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="b6">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="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">a9</font>,<font color="Olive" title="b5">b9</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">a9</font>,<font color="Olive" title="b6">c9</font> holds <br/><font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">b9</font>,<font color="Olive" title="b6">c9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">a9</font>, <font color="Maroon" title="c9">b9</font>, <font color="Maroon" title="c10">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c5">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="c8">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="c6">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="c9">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="c7">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="c10">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="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">c9</font> implies <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font> )</span><br/>

















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:21"/><span class="lab"><font color="Green" title="E17">A33</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E4:21"/><span class="lab"><font color="Green" title="E18">A34</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E5:21"/><span class="lab"><font color="Green" title="E19">A35</font></span>: 
<font color="Maroon" title="c5">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="E6:21"/><span class="lab"><font color="Green" title="E20">A36</font></span>: 
<font color="Maroon" title="c8">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="E7:21"/><span class="lab"><font color="Green" title="E21">A37</font></span>: 
<font color="Maroon" title="c6">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="E8:21"/><span class="lab"><font color="Green" title="E22">A38</font></span>: 
<font color="Maroon" title="c9">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="E9:21"/><span class="lab"><font color="Green" title="E23">A39</font></span>: 
<font color="Maroon" title="c7">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="E10:21"/><span class="lab"><font color="Green" title="E24">A40</font></span>: 
<font color="Maroon" title="c10">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="E11:21"/><span class="lab"><font color="Green" title="E25">A41</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="E12:21"/><span class="lab"><font color="Green" title="E26">A42</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="E13:21"/><span class="lab"><font color="Green" title="E27">A43</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="E14:21"/><span class="lab"><font color="Green" title="E28">A44</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="E15:21"/><span class="lab"><font color="Green" title="E29">A45</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="E16:21"/><span class="lab"><font color="Green" title="E30">A46</font></span>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font>
 <span class="kw">and </span><br/><a NAME="E17:21"/><span class="lab"><font color="Green" title="E31">A47</font></span>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font></span><br/>

<a NAME="E18:21"/><span class="lab"><font color="Green" title="E32">A48</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E17">A33</font></span></a>, <a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E19">A35</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E21">A37</font></span></a>, <a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E28">A44</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/>
<a NAME="E19:21"/><span class="lab"><font color="Green" title="E33">A49</font></span>: 
<font color="Maroon" title="c8">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E17">A33</font></span></a>, <a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E20">A36</font></span></a>, <a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E22">A38</font></span></a>, <a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E28">A44</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">K</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c12">N</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font>);<br/>
<a NAME="E20:21"/><span class="lab"><font color="Green" title="E34">A50</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E18">A34</font></span></a>, <a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E19">A35</font></span></a>, <a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E23">A39</font></span></a>, <a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E29">A45</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/>
<a NAME="E21:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A51</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<span class="kw">assume </span><a NAME="E22:21"/><span class="lab"><font color="Green" title="E36">A52</font></span>: 
 not <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E23:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A53</font></span>: 
<font color="Maroon" title="c9">b9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T3" title="AFF_1:th.3">AFF_1:3</a></span>;<br/>
<a NAME="E24:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A54</font></span>: 
<font color="Maroon" title="c9">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">b9</font>,<font color="Maroon" title="c10">c9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<a NAME="E25:21"/><span class="lab"><font color="Green" title="E39">A55</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:21"><span class="lab"><font color="Green" title="E36">A52</font></span></a>, <a class="ref" href="aff_1.html#T3" title="AFF_1:th.3">AFF_1:3</a></span>;<br/>
<a NAME="E26:21"/><span class="lab"><font color="Green" title="E40">A56</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_2"/><span class="lab"><font color="Green" title="E41">A57</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E3:21_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">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="E4:21_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E5:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A58</font></span>: 
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E30">A46</font></span></a>, <a class="txt" href="#E18:21"><span class="lab"><font color="Green" title="E32">A48</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E6:21_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2"><span class="lab"><font color="Green" title="E41">A57</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E7:21_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c5">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="E8:21_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E9:21_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E31">A47</font></span></a>, <a class="txt" href="#E20:21"><span class="lab"><font color="Green" title="E34">A50</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E10:21_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E39">A55</font></span></a>, <a class="txt" href="#E5:21_2"><span class="lab"><font color="Green" title="E42">A58</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E11:21_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c9">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="E12:21_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c8">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E13:21_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c8">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="E14:21_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:21_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:21"><span class="lab"><font color="Green" title="E36">A52</font></span></a>, <a class="txt" href="#E19:21"><span class="lab"><font color="Green" title="E33">A49</font></span></a>, <a class="txt" href="#E5:21_2"><span class="lab"><font color="Green" title="E42">A58</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E27:21"/><span class="lab"><font color="Green" title="E41">A59</font></span>: 
<font color="Maroon" title="c7">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="c5">a</font>,<font color="Maroon" title="c7">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:21"><span class="lab"><font color="Green" title="E34">A50</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<a NAME="E28:21"/><span class="lab"><font color="Green" title="E42">A60</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">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="#E23:21"><span class="lab"><font color="Green" title="E37">A53</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">M</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="E30:21"/><span class="lab"><font color="Green" title="E43">A61</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">M</font>
 <span class="kw">and </span><br/><a NAME="E31:21"/><span class="lab"><font color="Green" title="E44">A62</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c13">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T49" title="AFF_1:th.49">AFF_1:49</a></span>;<br/>
<a NAME="E32:21"/><span class="lab"><font color="Green" title="E45">A63</font></span>: 
<font color="Maroon" title="c10">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">b9</font>,<font color="Maroon" title="c10">c9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:21"><span class="lab"><font color="Green" title="E37">A53</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<a NAME="E33:21"/><span class="lab"><font color="Green" title="E46">A64</font></span>: 
<font color="Maroon" title="c8">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E18">A34</font></span></a>, <a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E20">A36</font></span></a>, <a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E24">A40</font></span></a>, <a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E29">A45</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/>
<a NAME="E34:21"/><span class="lab"><font color="Green" title="E47">A65</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_3"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">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="E3:21_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E30">A46</font></span></a>, <a class="txt" href="#E19:21"><span class="lab"><font color="Green" title="E33">A49</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E4:21_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E31">A47</font></span></a>, <a class="txt" href="#E33:21"><span class="lab"><font color="Green" title="E46">A64</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:21_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:21"><span class="lab"><font color="Green" title="E40">A56</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E35:21"/><span class="lab"><font color="Green" title="E48">A66</font></span>: 
 not  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c13">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_4"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c13">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_4"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:21"><span class="lab"><font color="Green" title="E44">A62</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/>
<a NAME="E3:21_4"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:21"><span class="lab"><font color="Green" title="E35">A51</font></span></a>, <a class="txt" href="#E27:21"><span class="lab"><font color="Green" title="E41">A59</font></span></a>, <a class="txt" href="#E24:21"><span class="lab"><font color="Green" title="E38">A54</font></span></a>, <a class="txt" href="#E32:21"><span class="lab"><font color="Green" title="E45">A63</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<a NAME="E4:21_4"/><span class="kw">then </span>
<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E31">A47</font></span></a>, <a class="txt" href="#E20:21"><span class="lab"><font color="Green" title="E34">A50</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E5:21_4"/><span class="kw">then </span>
<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c8">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c9">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E6:21_4"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">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/>
<span class="kw">hence </span><a NAME="E7:21_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:21"><span class="lab"><font color="Green" title="E47">A65</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E36:21"/><span class="lab"><font color="Green" title="E49">A67</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">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="#E20:21"><span class="lab"><font color="Green" title="E34">A50</font></span></a>, <a class="ref" href="aff_1.html#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<a NAME="E37:21"/><span class="lab"><font color="Green" title="E50">A68</font></span>: 
<font color="Maroon" title="c13">M</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="#E31:21"><span class="lab"><font color="Green" title="E44">A62</font></span></a>, <a class="ref" href="aff_1.html#T36" title="AFF_1:th.36">AFF_1:36</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">x</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="E39:21"/><span class="lab"><font color="Green" title="E51">A69</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>)
 <span class="kw">and </span><br/><a NAME="E40:21"/><span class="lab"><font color="Green" title="E52">A70</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:21"><span class="lab"><font color="Green" title="E49">A67</font></span></a>, <a class="txt" href="#E35:21"><span class="lab"><font color="Green" title="E48">A66</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<a NAME="E41:21"/><span class="lab"><font color="Green" title="E53">A71</font></span>: 
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c14">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:21"><span class="lab"><font color="Green" title="E38">A54</font></span></a>, <a class="txt" href="#E32:21"><span class="lab"><font color="Green" title="E45">A63</font></span></a>, <a class="txt" href="#E30:21"><span class="lab"><font color="Green" title="E43">A61</font></span></a>, <a class="txt" href="#E31:21"><span class="lab"><font color="Green" title="E44">A62</font></span></a>, <a class="txt" href="#E40:21"><span class="lab"><font color="Green" title="E52">A70</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="c15">D</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c10">c9</font>);<br/>
<a NAME="E42:21"/><span class="lab"><font color="Green" title="E54">A72</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c10">c9</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_5"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c10">c9</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_5"/><span class="kw">then </span>
<font color="Maroon" title="c10">c9</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="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:21_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E18">A34</font></span></a>, <a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E24">A40</font></span></a>, <a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E29">A45</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E43:21"/><span class="lab"><font color="Green" title="E55">A73</font></span>: 
<font color="Maroon" title="c14">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="c14">x</font>,<font color="Maroon" title="c10">c9</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="E44:21"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c14">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:21"><span class="lab"><font color="Green" title="E49">A67</font></span></a>, <a class="txt" href="#E21:21"><span class="lab"><font color="Green" title="E35">A51</font></span></a>, <a class="txt" href="#E27:21"><span class="lab"><font color="Green" title="E41">A59</font></span></a>, <a class="txt" href="#E39:21"><span class="lab"><font color="Green" title="E51">A69</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<a NAME="E45:21"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c14">x</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="E46:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A74</font></span>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c14">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E31">A47</font></span></a>, <a class="txt" href="#E20:21"><span class="lab"><font color="Green" title="E34">A50</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E47:21"/><span class="lab"><font color="Green" title="E57">A75</font></span>: 
<font color="Maroon" title="c10">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="c14">x</font>,<font color="Maroon" title="c10">c9</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="E48:21"/><span class="lab"><font color="Green" title="E58">A76</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c14">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_6"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:21_6"/><span class="lab"><font color="Green" title="E59">A77</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_6_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:21"><span class="lab"><font color="Green" title="E38">A54</font></span></a>, <a class="txt" href="#E32:21"><span class="lab"><font color="Green" title="E45">A63</font></span></a>, <a class="txt" href="#E30:21"><span class="lab"><font color="Green" title="E43">A61</font></span></a>, <a class="txt" href="#E31:21"><span class="lab"><font color="Green" title="E44">A62</font></span></a>, <a class="txt" href="#E40:21"><span class="lab"><font color="Green" title="E52">A70</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<a NAME="E3:21_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E30">A46</font></span></a>, <a class="txt" href="#E18:21"><span class="lab"><font color="Green" title="E32">A48</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E4:21_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c8">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E5:21_6_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">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/>
<span class="kw">hence </span><a NAME="E6:21_6_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:21"><span class="lab"><font color="Green" title="E47">A65</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E2:21_6"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c14">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:21_6"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E4:21_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A78</font></span>: 
<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c5">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="E5:21_6"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:21"><span class="lab"><font color="Green" title="E49">A67</font></span></a>, <a class="txt" href="#E21:21"><span class="lab"><font color="Green" title="E35">A51</font></span></a>, <a class="txt" href="#E27:21"><span class="lab"><font color="Green" title="E41">A59</font></span></a>, <a class="txt" href="#E26:21"><span class="lab"><font color="Green" title="E40">A56</font></span></a>, <a class="txt" href="#E39:21"><span class="lab"><font color="Green" title="E51">A69</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:21_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:21"><span class="lab"><font color="Green" title="E49">A67</font></span></a>, <a class="txt" href="#E21:21"><span class="lab"><font color="Green" title="E35">A51</font></span></a>, <a class="txt" href="#E30:21"><span class="lab"><font color="Green" title="E43">A61</font></span></a>, <a class="txt" href="#E37:21"><span class="lab"><font color="Green" title="E50">A68</font></span></a>, <a class="txt" href="#E35:21"><span class="lab"><font color="Green" title="E48">A66</font></span></a>, <a class="txt" href="#E39:21"><span class="lab"><font color="Green" title="E51">A69</font></span></a>, <a class="txt" href="#E40:21"><span class="lab"><font color="Green" title="E52">A70</font></span></a>, <a class="txt" href="#E4:21_6"><span class="lab"><font color="Green" title="E60">A78</font></span></a>, <a class="txt" href="#E1:21_6"><span class="lab"><font color="Green" title="E59">A77</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E49:21"/><span class="lab"><font color="Green" title="E59">A79</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E17">A33</font></span></a>, <a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E18">A34</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/>
<a NAME="E50:21"/><span class="lab"><font color="Green" title="E60">A80</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_7"><span class="kw">proof </span></a><div class="add">

<div><span class="lab"><font color="Green" title="E61">A81</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_7_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:21_7_1"/><span class="lab"><font color="Green" title="E62">A82</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E17">A33</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/><span class="kw">assume </span><a NAME="E2:21_7_1"/><span class="lab"><font color="Green" title="E63">A83</font></span>: 
<font color="Maroon" title="c3">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="c9">b9</font>,<font color="Maroon" title="c10">c9</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:21_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">c</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="#E9:21"><span class="lab"><font color="Green" title="E23">A39</font></span></a>, <a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E24">A40</font></span></a>, <a class="txt" href="#E49:21"><span class="lab"><font color="Green" title="E59">A79</font></span></a>, <a class="txt" href="#E32:21"><span class="lab"><font color="Green" title="E45">A63</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/><span class="kw">hence </span><a NAME="E4:21_7_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E21">A37</font></span></a>, <a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E22">A38</font></span></a>, <a class="txt" href="#E22:21"><span class="lab"><font color="Green" title="E36">A52</font></span></a>, <a class="txt" href="#E32:21"><span class="lab"><font color="Green" title="E45">A63</font></span></a>, <a class="txt" href="#E2:21_7_1"><span class="lab"><font color="Green" title="E63">A83</font></span></a>, <a class="txt" href="#E1:21_7_1"><span class="lab"><font color="Green" title="E62">A82</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E2:21_7"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:21_7"/><span class="kw">then </span>
<font color="Maroon" title="c13">M</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="c9">b9</font>,<font color="Maroon" title="c10">c9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:21"><span class="lab"><font color="Green" title="E45">A63</font></span></a>, <a class="txt" href="#E31:21"><span class="lab"><font color="Green" title="E44">A62</font></span></a>, <a class="txt" href="#E40:21"><span class="lab"><font color="Green" title="E52">A70</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/>
<a NAME="E4:21_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A84</font></span>: 
( <font color="Maroon" title="c3">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="c9">b9</font>,<font color="Maroon" title="c10">c9</font>) or <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E21">A37</font></span></a>, <a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E22">A38</font></span></a>, <a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E26">A42</font></span></a>, <a class="txt" href="#E28:21"><span class="lab"><font color="Green" title="E42">A60</font></span></a>, <a class="txt" href="#E24:21"><span class="lab"><font color="Green" title="E38">A54</font></span></a>, <a class="txt" href="#E30:21"><span class="lab"><font color="Green" title="E43">A61</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/>
<a NAME="E5:21_7"/><span class="kw">then </span>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E30">A46</font></span></a>, <a class="txt" href="#E1:21_7"><span class="lab"><font color="Green" title="E61">A81</font></span></a>, <a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E6:21_7"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">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="E7:21_7"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c6">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E8:21_7"/><span class="kw">then </span>
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E19">A35</font></span></a>, <a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E20">A36</font></span></a>, <a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E25">A41</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E9:21_7"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E17">A33</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E21">A37</font></span></a>, <a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E28">A44</font></span></a>, <a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E31">A47</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E10:21_7"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c8">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E11:21_7"/><span class="kw">then </span>
( <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c9</font> or <font color="Maroon" title="c8">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="#E9:21"><span class="lab"><font color="Green" title="E23">A39</font></span></a>, <a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E24">A40</font></span></a>, <a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E27">A43</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:21_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E18">A34</font></span></a>, <a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E20">A36</font></span></a>, <a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E29">A45</font></span></a>, <a class="txt" href="#E22:21"><span class="lab"><font color="Green" title="E36">A52</font></span></a>, <a class="txt" href="#E4:21_7"><span class="lab"><font color="Green" title="E62">A84</font></span></a>, <a class="txt" href="#E1:21_7"><span class="lab"><font color="Green" title="E61">A81</font></span></a>, <a class="ref" href="aff_1.html#T2" title="AFF_1:th.2">AFF_1:2</a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E51:21"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c10">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#T24" title="AFF_1:th.24">AFF_1:24</a></span>;<br/>
<a NAME="E52:21"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c10">c9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E16">A32</font></span></a>, <a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E17">A33</font></span></a>, <a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E19">A35</font></span></a>, <a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E20">A36</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E21">A37</font></span></a>, <a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E22">A38</font></span></a>, <a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E25">A41</font></span></a>, <a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E26">A42</font></span></a>, <a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E28">A44</font></span></a>, <a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E30">A46</font></span></a>, <a class="txt" href="#E41:21"><span class="lab"><font color="Green" title="E53">A71</font></span></a>, <a class="txt" href="#E46:21"><span class="lab"><font color="Green" title="E56">A74</font></span></a>, <a class="txt" href="#E43:21"><span class="lab"><font color="Green" title="E55">A73</font></span></a>, <a class="txt" href="#E47:21"><span class="lab"><font color="Green" title="E57">A75</font></span></a>, <a class="txt" href="#E50:21"><span class="lab"><font color="Green" title="E60">A80</font></span></a>, <a class="txt" href="#E48:21"><span class="lab"><font color="Green" title="E58">A76</font></span></a>, <a class="txt" href="#E42:21"><span class="lab"><font color="Green" title="E54">A72</font></span></a>, <a class="ref" href="aff_2.html#D12" target="_self" title="AFF_2:def.12">Def12</a></span>;<br/>
<a NAME="E53:21"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c10">c9</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E18">A34</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/>
<a NAME="E54:21"/><span class="kw">then </span>
<font color="Maroon" title="c4">C</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="c14">x</font>,<font color="Maroon" title="c10">c9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E24">A40</font></span></a>, <a class="txt" href="#E47:21"><span class="lab"><font color="Green" title="E57">A75</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/>
<a NAME="E55:21"/><span class="kw">then </span>
<font color="Maroon" title="c4">C</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="c5">a</font>,<font color="Maroon" title="c7">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E23">A39</font></span></a>, <a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E27">A43</font></span></a>, <a class="txt" href="#E22:21"><span class="lab"><font color="Green" title="E36">A52</font></span></a>, <a class="txt" href="#E36:21"><span class="lab"><font color="Green" title="E49">A67</font></span></a>, <a class="txt" href="#E27:21"><span class="lab"><font color="Green" title="E41">A59</font></span></a>, <a class="txt" href="#E39:21"><span class="lab"><font color="Green" title="E51">A69</font></span></a>, <a class="txt" href="#E41:21"><span class="lab"><font color="Green" title="E53">A71</font></span></a>, <a class="txt" href="#E43:21"><span class="lab"><font color="Green" title="E55">A73</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E56:21"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E18">A34</font></span></a>, <a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E19">A35</font></span></a>, <a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E29">A45</font></span></a>, <a class="txt" href="#E21:21"><span class="lab"><font color="Green" title="E35">A51</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
