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

<span class="kw">let </span><font color="Maroon" title="c1">AP</font> be   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V13" title="AFF_2:attr.13">satisfying_pap</a>  iff <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V14" title="AFF_2:attr.14">satisfying_pap_1</a>  )</span><br/>

<div><span class="lab"><font color="Green" title="E16">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:36_1"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V13" title="AFF_2:attr.13">satisfying_pap</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V14" title="AFF_2:attr.14">satisfying_pap_1</a> </span><br/><span class="kw">thus </span><a NAME="E2:36_1"/>
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V14" title="AFF_2:attr.14">satisfying_pap_1</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">M</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="aff_2.html#D14" target="_self" title="AFF_2:def.14">AFF_2:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">N</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="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font>, <font color="Olive" title="b5">a9</font>, <font color="Olive" title="b6">b9</font>, <font color="Olive" title="b7">c9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>  st <font color="Maroon" title="c2">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c2">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b1">N</font> &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b5">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b6">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b6">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b5">a9</font> &amp; <font color="Olive" title="b3">b</font>,<font color="Olive" title="b7">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">c</font>,<font color="Olive" title="b6">b9</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b7">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">c</font>,<font color="Olive" title="b5">a9</font> &amp; <font color="Olive" title="b5">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">b9</font> holds <br/><font color="Olive" title="b7">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">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">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c2">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b4">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</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">N</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">a9</font> &amp; <font color="Olive" title="b2">b</font>,<font color="Olive" title="b6">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b5">b9</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b6">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">a9</font> &amp; <font color="Olive" title="b4">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">b9</font> holds <br/><font color="Olive" title="b6">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">a9</font>, <font color="Maroon" title="c8">b9</font>, <font color="Maroon" title="c9">c9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c2">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c7">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">a9</font> &amp; <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">a9</font> &amp; <font color="Maroon" title="c7">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b9</font> implies <font color="Maroon" title="c9">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> )</span><br/>















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:36_1_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c2">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E2:36_1_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<font color="Maroon" title="c3">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E3:36_1_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">and </span><br/><a NAME="E4:36_1_1"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">and </span><br/><a NAME="E5:36_1_1"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">and </span><br/><a NAME="E6:36_1_1"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
<font color="Maroon" title="c2">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E7:36_1_1"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
<font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E8:36_1_1"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
<font color="Maroon" title="c7">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E9:36_1_1"/><span class="lab"><font color="Green" title="E26">A11</font></span>: 
<font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E10:36_1_1"/><span class="lab"><font color="Green" title="E27">A12</font></span>: 
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">a9</font>
 <span class="kw">and </span><br/><a NAME="E11:36_1_1"/><span class="lab"><font color="Green" title="E28">A13</font></span>: 
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font>
 <span class="kw">and </span><br/><a NAME="E12:36_1_1"/><span class="lab"><font color="Green" title="E29">A14</font></span>: 
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">a9</font>
 <span class="kw">and </span><br/><a NAME="E13:36_1_1"/><span class="lab"><font color="Green" title="E30">A15</font></span>: 
<font color="Maroon" title="c7">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font></span><br/>

<a NAME="E14:36_1_1"/><span class="lab"><font color="Green" title="E31">A16</font></span>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E6:36_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E7:36_1_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E8:36_1_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">C</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font>;<br/>
<a NAME="E15:36_1_1"/><span class="lab"><font color="Green" title="E32">A17</font></span>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E6:36_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E7:36_1_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E9:36_1_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E16:36_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<span class="kw">then </span><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="E18:36_1_1"/><span class="lab"><font color="Green" title="E33">A18</font></span>: 
<font color="Maroon" title="c5">b</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="E19:36_1_1"/><span class="lab"><font color="Green" title="E34">A19</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c11">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></span>;<br/>
<a NAME="E20:36_1_1"/><span class="lab"><font color="Green" title="E35">A20</font></span>: 
( <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font> &amp; <font color="Maroon" title="c8">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="c6">c</font>,<font color="Maroon" title="c8">b9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:36_1_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E21:36_1_1"/><span class="lab"><font color="Green" title="E36">A21</font></span>: 
not <font color="Maroon" title="c11">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:36_1_1_1"/>
<font color="Maroon" title="c11">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:36_1_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:36_1_1"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<a NAME="E3:36_1_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:36_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<a NAME="E4:36_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E20:36_1_1"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:36_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:36_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E7:36_1_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E9:36_1_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E22:36_1_1"/>
<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="#E19:36_1_1"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="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="E24:36_1_1"/><span class="lab"><font color="Green" title="E37">A22</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="E25:36_1_1"/><span class="lab"><font color="Green" title="E38">A23</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E21:36_1_1"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></span>;<br/>
<a NAME="E26:36_1_1"/><span class="lab"><font color="Green" title="E39">A24</font></span>: 
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c12">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36_1_1"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E18:36_1_1"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E19:36_1_1"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="txt" href="#E24:36_1_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/>
<a NAME="E27:36_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c12">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E1:36_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E2:36_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E3:36_1_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E4:36_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E5:36_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E6:36_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E7:36_1_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E8:36_1_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E9:36_1_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E10:36_1_1"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E25:36_1_1"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="ref" href="aff_2.html#D13" target="_self" title="AFF_2:def.13">Def13</a></span>;<br/>
<a NAME="E28:36_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c12">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:36_1_1"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="txt" href="#E14:36_1_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E29:36_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c9">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="E30:36_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A25</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c9</font>,<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c4">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E31:36_1_1"/><span class="lab"><font color="Green" title="E41">A26</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:36_1_1_2"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:36_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="c4">a</font>,<font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c7">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:36_1_1"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E3:36_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="c7">a9</font>,<font color="Maroon" title="c8">b9</font>,<font color="Maroon" title="c4">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E4:36_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c7">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b9</font> or <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E8:36_1_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E9:36_1_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:36_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36_1_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E6:36_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E7:36_1_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E13:36_1_1"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E32:36_1_1"/><span class="lab"><font color="Green" title="E42">A27</font></span>: 
<font color="Maroon" title="c9">c9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:36_1_1_3"/>
<font color="Maroon" title="c9">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:36_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E3:36_1_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E4:36_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E5:36_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E12:36_1_1"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="txt" href="#E31:36_1_1"><span class="lab"><font color="Green" title="E41">A26</font></span></a>, <a class="ref" href="aff_1.html#T62" title="AFF_1:th.62">AFF_1:62</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:36_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:36_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E7:36_1_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E8:36_1_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E33:36_1_1"/>
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c12">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c9">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_1_1"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E15:36_1_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E26:36_1_1"><span class="lab"><font color="Green" title="E39">A24</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E34:36_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c9">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="E35:36_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A28</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c9</font>,<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c5">b</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">assume </span><a NAME="E36:36_1_1"/><span class="lab"><font color="Green" title="E44">A29</font></span>: 
not <font color="Maroon" title="c9">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E37:36_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c9</font>,<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c9">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E38:36_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E3:36_1_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E4:36_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E36:36_1_1"><span class="lab"><font color="Green" title="E44">A29</font></span></a>, <a class="txt" href="#E31:36_1_1"><span class="lab"><font color="Green" title="E41">A26</font></span></a>, <a class="txt" href="#E25:36_1_1"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="txt" href="#E30:36_1_1"><span class="lab"><font color="Green" title="E40">A25</font></span></a>, <a class="txt" href="#E35:36_1_1"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<a NAME="E39:36_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E4:36_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E5:36_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E11:36_1_1"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E32:36_1_1"><span class="lab"><font color="Green" title="E42">A27</font></span></a>, <a class="ref" href="aff_1.html#T62" title="AFF_1:th.62">AFF_1:62</a></span>;<br/>
<span class="kw">hence </span><a NAME="E40:36_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:36_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E7:36_1_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E9:36_1_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:36_2"/><span class="lab"><font color="Green" title="E17">A30</font></span>: 
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V14" title="AFF_2:attr.14">satisfying_pap_1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V13" title="AFF_2:attr.13">satisfying_pap</a> </span><br/><span class="kw">thus </span><a NAME="E2:36_2"/>
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V13" title="AFF_2:attr.13">satisfying_pap</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">M</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="aff_2.html#D13" target="_self" title="AFF_2:def.13">AFF_2:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">N</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="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font>, <font color="Olive" title="b5">a9</font>, <font color="Olive" title="b6">b9</font>, <font color="Olive" title="b7">c9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>  st <font color="Maroon" title="c2">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c2">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b1">N</font> &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b5">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b6">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b7">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b6">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b5">a9</font> &amp; <font color="Olive" title="b3">b</font>,<font color="Olive" title="b7">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">c</font>,<font color="Olive" title="b6">b9</font> holds <br/><font color="Olive" title="b2">a</font>,<font color="Olive" title="b7">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">c</font>,<font color="Olive" title="b5">a9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">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">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c2">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b4">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</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">N</font> &amp; <font color="Olive" title="b6">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">a9</font> &amp; <font color="Olive" title="b2">b</font>,<font color="Olive" title="b6">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b5">b9</font> holds <br/><font color="Olive" title="b1">a</font>,<font color="Olive" title="b6">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">a9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">a9</font>, <font color="Maroon" title="c8">b9</font>, <font color="Maroon" title="c9">c9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c2">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c7">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c9">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">a9</font> &amp; <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font> implies <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">a9</font> )</span><br/>















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:36_2_1"/><span class="lab"><font color="Green" title="E18">A31</font></span>: 
<font color="Maroon" title="c2">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E2:36_2_1"/><span class="lab"><font color="Green" title="E19">A32</font></span>: 
<font color="Maroon" title="c3">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E3:36_2_1"/><span class="lab"><font color="Green" title="E20">A33</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">and </span><br/><a NAME="E4:36_2_1"/><span class="lab"><font color="Green" title="E21">A34</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">and </span><br/><a NAME="E5:36_2_1"/><span class="lab"><font color="Green" title="E22">A35</font></span>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">and </span><br/><a NAME="E6:36_2_1"/><span class="lab"><font color="Green" title="E23">A36</font></span>: 
( <font color="Maroon" title="c2">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font> )
 <span class="kw">and </span><br/><a NAME="E7:36_2_1"/><span class="lab"><font color="Green" title="E24">A37</font></span>: 
<font color="Maroon" title="c7">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E8:36_2_1"/><span class="lab"><font color="Green" title="E25">A38</font></span>: 
<font color="Maroon" title="c8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E9:36_2_1"/><span class="lab"><font color="Green" title="E26">A39</font></span>: 
<font color="Maroon" title="c9">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E10:36_2_1"/><span class="lab"><font color="Green" title="E27">A40</font></span>: 
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">a9</font>
 <span class="kw">and </span><br/><a NAME="E11:36_2_1"/><span class="lab"><font color="Green" title="E28">A41</font></span>: 
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">a9</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c10">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">a9</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">C</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">D</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c9">c9</font>;<br/>
<a NAME="E12:36_2_1"/><span class="lab"><font color="Green" title="E29">A42</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_2_1"><span class="lab"><font color="Green" title="E21">A34</font></span></a>, <a class="txt" href="#E6:36_2_1"><span class="lab"><font color="Green" title="E23">A36</font></span></a>, <a class="txt" href="#E9:36_2_1"><span class="lab"><font color="Green" title="E26">A39</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E13:36_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A43</font></span>: 
( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c9">c9</font> &amp; <font color="Maroon" title="c9">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="c5">b</font>,<font color="Maroon" title="c9">c9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<span class="kw">assume </span><a NAME="E14:36_2_1"/><span class="lab"><font color="Green" title="E31">A44</font></span>: 
not <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">a9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E15:36_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A45</font></span>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T12" title="AFF_1:th.12">AFF_1:12</a></span>;<br/>
<a NAME="E16:36_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A46</font></span>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E17:36_2_1"/><span class="lab"><font color="Green" title="E34">A47</font></span>: 
<font color="Maroon" title="c7">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:36_2_1"><span class="lab"><font color="Green" title="E32">A45</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E18:36_2_1"/><span class="lab"><font color="Green" title="E35">A48</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">a9</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="#E15:36_2_1"><span class="lab"><font color="Green" title="E32">A45</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">P</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="E20:36_2_1"/><span class="lab"><font color="Green" title="E36">A49</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P</font>
 <span class="kw">and </span><br/><a NAME="E21:36_2_1"/><span class="lab"><font color="Green" title="E37">A50</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">a9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c13">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></span>;<br/>
<a NAME="E22:36_2_1"/><span class="lab"><font color="Green" title="E38">A51</font></span>: 
<font color="Maroon" title="c7">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:36_2_1_1"/><span class="lab"><font color="Green" title="E39">A52</font></span>: 
<font color="Maroon" title="c7">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:36_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c4">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c5">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:36_2_1"><span class="lab"><font color="Green" title="E27">A40</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E3:36_2_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E4:36_2_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">a9</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="E5:36_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> or <font color="Maroon" title="c7">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_2_1"><span class="lab"><font color="Green" title="E18">A31</font></span></a>, <a class="txt" href="#E3:36_2_1"><span class="lab"><font color="Green" title="E20">A33</font></span></a>, <a class="txt" href="#E4:36_2_1"><span class="lab"><font color="Green" title="E21">A34</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:36_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:36_2_1"><span class="lab"><font color="Green" title="E23">A36</font></span></a>, <a class="txt" href="#E7:36_2_1"><span class="lab"><font color="Green" title="E24">A37</font></span></a>, <a class="txt" href="#E11:36_2_1"><span class="lab"><font color="Green" title="E28">A41</font></span></a>, <a class="txt" href="#E14:36_2_1"><span class="lab"><font color="Green" title="E31">A44</font></span></a>, <a class="txt" href="#E1:36_2_1_1"><span class="lab"><font color="Green" title="E39">A52</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E23:36_2_1"/><span class="lab"><font color="Green" title="E39">A53</font></span>: 
not  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c9">c9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c13">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:36_2_1_2"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c9">c9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c13">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:36_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c13">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_2_1"><span class="lab"><font color="Green" title="E28">A41</font></span></a>, <a class="txt" href="#E12:36_2_1"><span class="lab"><font color="Green" title="E29">A42</font></span></a>, <a class="txt" href="#E13:36_2_1"><span class="lab"><font color="Green" title="E30">A43</font></span></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></span>;<br/>
<a NAME="E3:36_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:36_2_1"><span class="lab"><font color="Green" title="E37">A50</font></span></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></span>;<br/>
<a NAME="E4:36_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">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="c6">c</font>,<font color="Maroon" title="c7">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:36_2_1"><span class="lab"><font color="Green" title="E35">A48</font></span></a>, <a class="txt" href="#E16:36_2_1"><span class="lab"><font color="Green" title="E33">A46</font></span></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></span>;<br/>
<a NAME="E5:36_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_2_1"><span class="lab"><font color="Green" title="E19">A32</font></span></a>, <a class="txt" href="#E7:36_2_1"><span class="lab"><font color="Green" title="E24">A37</font></span></a>, <a class="txt" href="#E8:36_2_1"><span class="lab"><font color="Green" title="E25">A38</font></span></a>, <a class="txt" href="#E22:36_2_1"><span class="lab"><font color="Green" title="E38">A51</font></span></a>, <a class="txt" href="#E18:36_2_1"><span class="lab"><font color="Green" title="E35">A48</font></span></a>, <a class="txt" href="#E16:36_2_1"><span class="lab"><font color="Green" title="E33">A46</font></span></a>, <a class="txt" href="#E17:36_2_1"><span class="lab"><font color="Green" title="E34">A47</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:36_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_2_1"><span class="lab"><font color="Green" title="E22">A35</font></span></a>, <a class="txt" href="#E6:36_2_1"><span class="lab"><font color="Green" title="E23">A36</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E24:36_2_1"/><span class="lab"><font color="Green" title="E40">A54</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c9">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="#E12:36_2_1"><span class="lab"><font color="Green" title="E29">A42</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E25:36_2_1"/>
<font color="Maroon" title="c13">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:36_2_1"><span class="lab"><font color="Green" title="E37">A50</font></span></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="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="E27:36_2_1"/><span class="lab"><font color="Green" title="E41">A55</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">b</font>,<font color="Maroon" title="c9">c9</font>
 <span class="kw">and </span><br/><a NAME="E28:36_2_1"/><span class="lab"><font color="Green" title="E42">A56</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">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:36_2_1"><span class="lab"><font color="Green" title="E40">A54</font></span></a>, <a class="txt" href="#E23:36_2_1"><span class="lab"><font color="Green" title="E39">A53</font></span></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></span>;<br/>
<a NAME="E29:36_2_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c9">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:36_2_1"><span class="lab"><font color="Green" title="E40">A54</font></span></a>, <a class="txt" href="#E13:36_2_1"><span class="lab"><font color="Green" title="E30">A43</font></span></a>, <a class="txt" href="#E27:36_2_1"><span class="lab"><font color="Green" title="E41">A55</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/>
<a NAME="E30:36_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c14">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c9">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="E31:36_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A57</font></span>: 
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c14">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_2_1"><span class="lab"><font color="Green" title="E28">A41</font></span></a>, <a class="txt" href="#E12:36_2_1"><span class="lab"><font color="Green" title="E29">A42</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E32:36_2_1"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c14">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:36_2_1"><span class="lab"><font color="Green" title="E33">A46</font></span></a>, <a class="txt" href="#E17:36_2_1"><span class="lab"><font color="Green" title="E34">A47</font></span></a>, <a class="txt" href="#E20:36_2_1"><span class="lab"><font color="Green" title="E36">A49</font></span></a>, <a class="txt" href="#E21:36_2_1"><span class="lab"><font color="Green" title="E37">A50</font></span></a>, <a class="txt" href="#E28:36_2_1"><span class="lab"><font color="Green" title="E42">A56</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/>
<a NAME="E33:36_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_2"><span class="lab"><font color="Green" title="E17">A30</font></span></a>, <a class="txt" href="#E1:36_2_1"><span class="lab"><font color="Green" title="E18">A31</font></span></a>, <a class="txt" href="#E2:36_2_1"><span class="lab"><font color="Green" title="E19">A32</font></span></a>, <a class="txt" href="#E3:36_2_1"><span class="lab"><font color="Green" title="E20">A33</font></span></a>, <a class="txt" href="#E4:36_2_1"><span class="lab"><font color="Green" title="E21">A34</font></span></a>, <a class="txt" href="#E5:36_2_1"><span class="lab"><font color="Green" title="E22">A35</font></span></a>, <a class="txt" href="#E6:36_2_1"><span class="lab"><font color="Green" title="E23">A36</font></span></a>, <a class="txt" href="#E7:36_2_1"><span class="lab"><font color="Green" title="E24">A37</font></span></a>, <a class="txt" href="#E8:36_2_1"><span class="lab"><font color="Green" title="E25">A38</font></span></a>, <a class="txt" href="#E10:36_2_1"><span class="lab"><font color="Green" title="E27">A40</font></span></a>, <a class="txt" href="#E22:36_2_1"><span class="lab"><font color="Green" title="E38">A51</font></span></a>, <a class="txt" href="#E31:36_2_1"><span class="lab"><font color="Green" title="E43">A57</font></span></a>, <a class="ref" href="aff_2.html#D14" target="_self" title="AFF_2:def.14">Def14</a></span>;<br/>
<a NAME="E34:36_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c9</font> or <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_2_1"><span class="lab"><font color="Green" title="E19">A32</font></span></a>, <a class="txt" href="#E9:36_2_1"><span class="lab"><font color="Green" title="E26">A39</font></span></a>, <a class="txt" href="#E24:36_2_1"><span class="lab"><font color="Green" title="E40">A54</font></span></a>, <a class="txt" href="#E13:36_2_1"><span class="lab"><font color="Green" title="E30">A43</font></span></a>, <a class="txt" href="#E27:36_2_1"><span class="lab"><font color="Green" title="E41">A55</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E35:36_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_2_1"><span class="lab"><font color="Green" title="E21">A34</font></span></a>, <a class="txt" href="#E6:36_2_1"><span class="lab"><font color="Green" title="E23">A36</font></span></a>, <a class="txt" href="#E14:36_2_1"><span class="lab"><font color="Green" title="E31">A44</font></span></a>, <a class="txt" href="#E16:36_2_1"><span class="lab"><font color="Green" title="E33">A46</font></span></a>, <a class="txt" href="#E17:36_2_1"><span class="lab"><font color="Green" title="E34">A47</font></span></a>, <a class="txt" href="#E20:36_2_1"><span class="lab"><font color="Green" title="E36">A49</font></span></a>, <a class="txt" href="#E21:36_2_1"><span class="lab"><font color="Green" title="E37">A50</font></span></a>, <a class="txt" href="#E28:36_2_1"><span class="lab"><font color="Green" title="E42">A56</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:36"/>
( <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V13" title="AFF_2:attr.13">satisfying_pap</a>  iff <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V14" title="AFF_2:attr.14">satisfying_pap_1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E16">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
