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<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#V4" title="AFF_2:attr.4">Desarguesian</a>  iff <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V5" title="AFF_2:attr.5">satisfying_DES_1</a>  )</span><br/>

<div><span class="lab"><font color="Green" title="E14">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:30_1"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V5" title="AFF_2:attr.5">satisfying_DES_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#V4" title="AFF_2:attr.4">Desarguesian</a> </span><br/><span class="kw">thus </span><a NAME="E2:30_1"/>
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_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="Red">:: according to </font></span><a class="ref" href="aff_2.html#D4" target="_self" title="AFF_2:def.4">AFF_2:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><br/> for <font color="Olive" title="b3">o</font>, <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font>, <font color="Olive" title="b6">c</font>, <font color="Olive" title="b7">a9</font>, <font color="Olive" title="b8">b9</font>, <font color="Olive" title="b9">c9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>  st <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">a</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">b</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">c</font> &amp; <font color="Olive" title="b4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b7">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b9">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp; <font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b4">a</font>,<font color="Olive" title="b5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">a9</font>,<font color="Olive" title="b8">b9</font> &amp; <font color="Olive" title="b4">a</font>,<font color="Olive" title="b6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">a9</font>,<font color="Olive" title="b9">c9</font> holds <br/><font color="Olive" title="b5">b</font>,<font color="Olive" title="b6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b8">b9</font>,<font color="Olive" title="b9">c9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font>, <font color="Olive" title="b5">a9</font>, <font color="Olive" title="b6">b9</font>, <font color="Olive" title="b7">c9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>  st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">c</font> &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b5">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b6">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b7">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">a9</font>,<font color="Olive" title="b6">b9</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">a9</font>,<font color="Olive" title="b7">c9</font> holds <br/><font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">b9</font>,<font color="Olive" title="b7">c9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">o</font>, <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c9">a9</font>, <font color="Maroon" title="c10">b9</font>, <font color="Maroon" title="c11">c9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c10">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c11">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> &amp; <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font> implies <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> )</span><br/>



















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:30_1_1"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E2:30_1_1"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E3:30_1_1"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E4:30_1_1"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">and </span><br/><a NAME="E5:30_1_1"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">and </span><br/><a NAME="E6:30_1_1"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c</font>
 <span class="kw">and </span><br/><a NAME="E7:30_1_1"/><span class="lab"><font color="Green" title="E22">A9</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E8:30_1_1"/><span class="lab"><font color="Green" title="E23">A10</font></span>: 
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E9:30_1_1"/><span class="lab"><font color="Green" title="E24">A11</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E10:30_1_1"/><span class="lab"><font color="Green" title="E25">A12</font></span>: 
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E11:30_1_1"/><span class="lab"><font color="Green" title="E26">A13</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E12:30_1_1"/><span class="lab"><font color="Green" title="E27">A14</font></span>: 
<font color="Maroon" title="c11">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E13:30_1_1"/><span class="lab"><font color="Green" title="E28">A15</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E14:30_1_1"/><span class="lab"><font color="Green" title="E29">A16</font></span>: 
<font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E15:30_1_1"/><span class="lab"><font color="Green" title="E30">A17</font></span>: 
<font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E16:30_1_1"/><span class="lab"><font color="Green" title="E31">A18</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E17:30_1_1"/><span class="lab"><font color="Green" title="E32">A19</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E18:30_1_1"/><span class="lab"><font color="Green" title="E33">A20</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font>
 <span class="kw">and </span><br/><a NAME="E19:30_1_1"/><span class="lab"><font color="Green" title="E34">A21</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font></span><br/>

<span class="kw">assume </span><a NAME="E20:30_1_1"/><span class="lab"><font color="Green" title="E35">A22</font></span>: 
not <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E21:30_1_1"/><span class="lab"><font color="Green" title="E36">A23</font></span>: 
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:30_1_1_1"/><span class="lab"><font color="Green" title="E37">A24</font></span>: 
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:30_1_1"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:30_1_1_1"/><span class="lab"><font color="Green" title="E38">A25</font></span>: 
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:30_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E2:30_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E8:30_1_1"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="txt" href="#E10:30_1_1"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="txt" href="#E13:30_1_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E14:30_1_1"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E16:30_1_1"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E4:30_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> or <font color="Maroon" title="c9">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_1"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="txt" href="#E12:30_1_1"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="#E15:30_1_1"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="txt" href="#E1:30_1_1_1"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="ref" href="aff_1.html#T62" title="AFF_1:th.62">AFF_1:62</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:30_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E4:30_1_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E7:30_1_1"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E13:30_1_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E15:30_1_1"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="txt" href="#E17:30_1_1"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="txt" href="#E20:30_1_1"><span class="lab"><font color="Green" title="E35">A22</font></span></a>, <a class="txt" href="#E2:30_1_1_1"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="ref" href="aff_1.html#T12" title="AFF_1:th.12">AFF_1:12</a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E22:30_1_1"/><span class="lab"><font color="Green" title="E37">A26</font></span>: 
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_1_1_2"/>
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:30_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A27</font></span>: 
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E2:30_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E8:30_1_1"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="txt" href="#E12:30_1_1"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="#E13:30_1_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E15:30_1_1"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="txt" href="#E17:30_1_1"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E3:30_1_1_2"/>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:30_1_1"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E4:30_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_1"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="txt" href="#E10:30_1_1"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="txt" href="#E14:30_1_1"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E21:30_1_1"><span class="lab"><font color="Green" title="E36">A23</font></span></a>, <a class="txt" href="#E2:30_1_1_2"><span class="lab"><font color="Green" title="E38">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="E5:30_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E2:30_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E4:30_1_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E7:30_1_1"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E13:30_1_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E14:30_1_1"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E16:30_1_1"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E23:30_1_1"/><span class="lab"><font color="Green" title="E38">A28</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_1_1_3"/><span class="lab"><font color="Green" title="E39">A29</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:30_1_1_3"/>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:30_1_1"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E3:30_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">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="#E1:30_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E7:30_1_1"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E8:30_1_1"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="txt" href="#E13:30_1_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E22:30_1_1"><span class="lab"><font color="Green" title="E37">A26</font></span></a>, <a class="txt" href="#E1:30_1_1_3"><span class="lab"><font color="Green" title="E39">A29</font></span></a>, <a class="ref" href="aff_1.html#T62" title="AFF_1:th.62">AFF_1:62</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:30_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E6:30_1_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E11:30_1_1"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="txt" href="#E13:30_1_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E15:30_1_1"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="txt" href="#E17:30_1_1"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c12">K</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">M</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">N</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>;<br/>
<a NAME="E24:30_1_1"/><span class="lab"><font color="Green" title="E39">A30</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E4:30_1_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E7:30_1_1"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E11:30_1_1"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="txt" href="#E13:30_1_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E15:30_1_1"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="txt" href="#E17:30_1_1"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E25:30_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A31</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</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="E26:30_1_1"/><span class="lab"><font color="Green" title="E41">A32</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E2:30_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E4:30_1_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E7:30_1_1"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E9:30_1_1"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="txt" href="#E13:30_1_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E14:30_1_1"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E16:30_1_1"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E27:30_1_1"/><span class="lab"><font color="Green" title="E42">A33</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_1_1_4"/><span class="lab"><font color="Green" title="E43">A34</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:30_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E3:30_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:30_1_1"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="txt" href="#E22:30_1_1"><span class="lab"><font color="Green" title="E37">A26</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E4:30_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:30_1_1"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="txt" href="#E21:30_1_1"><span class="lab"><font color="Green" title="E36">A23</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E5:30_1_1_4"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E6:30_1_1_4"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E7:30_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E8:30_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E9:30_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A35</font></span>: 
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:30_1_1"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="txt" href="#E26:30_1_1"><span class="lab"><font color="Green" title="E41">A32</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E10:30_1_1_4"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c9">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1_4"><span class="lab"><font color="Green" title="E43">A34</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E11:30_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c9">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E12:30_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:30_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:30_1_1"><span class="lab"><font color="Green" title="E35">A22</font></span></a>, <a class="txt" href="#E21:30_1_1"><span class="lab"><font color="Green" title="E36">A23</font></span></a>, <a class="txt" href="#E9:30_1_1_4"><span class="lab"><font color="Green" title="E44">A35</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E28:30_1_1"/><span class="lab"><font color="Green" title="E43">A36</font></span>: 
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:30_1_1"><span class="lab"><font color="Green" title="E35">A22</font></span></a>, <a class="ref" href="aff_1.html#T12" title="AFF_1:th.12">AFF_1:12</a></span>;<br/>
<a NAME="E29:30_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A37</font></span>: 
( <font color="Maroon" title="c10">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> &amp; <font color="Maroon" title="c11">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E30:30_1_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:30_1_1"><span class="lab"><font color="Green" title="E43">A36</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">D</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="E32:30_1_1"/><span class="lab"><font color="Green" title="E45">A38</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">D</font>
 <span class="kw">and </span><br/><a NAME="E33:30_1_1"/><span class="lab"><font color="Green" title="E46">A39</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c15">D</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="E34:30_1_1"/><span class="lab"><font color="Green" title="E47">A40</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:30_1_1"><span class="lab"><font color="Green" title="E39">A30</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E35:30_1_1"/><span class="lab"><font color="Green" title="E48">A41</font></span>: 
not  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c15">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_1_1_5"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c15">D</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="E2:30_1_1_5"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:30_1_1"><span class="lab"><font color="Green" title="E46">A39</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<a NAME="E3:30_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:30_1_1"><span class="lab"><font color="Green" title="E47">A40</font></span></a>, <a class="txt" href="#E25:30_1_1"><span class="lab"><font color="Green" title="E40">A31</font></span></a>, <a class="txt" href="#E29:30_1_1"><span class="lab"><font color="Green" title="E44">A37</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/>
<a NAME="E4:30_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:30_1_1"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="txt" href="#E24:30_1_1"><span class="lab"><font color="Green" title="E39">A30</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E5:30_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c9">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E6:30_1_1_5"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:30_1_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:30_1_1"><span class="lab"><font color="Green" title="E42">A33</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</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="E36:30_1_1"/><span class="lab"><font color="Green" title="E49">A42</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:30_1_1"><span class="lab"><font color="Green" title="E39">A30</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E37:30_1_1"/>
<font color="Maroon" title="c15">D</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:30_1_1"><span class="lab"><font color="Green" title="E46">A39</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="c16">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="E39:30_1_1"/><span class="lab"><font color="Green" title="E50">A43</font></span>: 
<font color="Maroon" title="c16">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="c6">a</font>,<font color="Maroon" title="c8">c</font>
 <span class="kw">and </span><br/><a NAME="E40:30_1_1"/><span class="lab"><font color="Green" title="E51">A44</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:30_1_1"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E35:30_1_1"><span class="lab"><font color="Green" title="E48">A41</font></span></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></span>;<br/>
<a NAME="E41:30_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c16">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:30_1_1"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E34:30_1_1"><span class="lab"><font color="Green" title="E47">A40</font></span></a>, <a class="txt" href="#E25:30_1_1"><span class="lab"><font color="Green" title="E40">A31</font></span></a>, <a class="txt" href="#E39:30_1_1"><span class="lab"><font color="Green" title="E50">A43</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/>
<a NAME="E42:30_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c16">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="E43:30_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A45</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c16">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:30_1_1"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="txt" href="#E24:30_1_1"><span class="lab"><font color="Green" title="E39">A30</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">T</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c16">x</font>;<br/>
<a NAME="E44:30_1_1"/><span class="lab"><font color="Green" title="E53">A46</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_1_1_6"/><span class="lab"><font color="Green" title="E54">A47</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:30_1_1_6"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:30_1_1"><span class="lab"><font color="Green" title="E34">A21</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="E3:30_1_1_6"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E4:30_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A48</font></span>: 
( <font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c9</font> or <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_1"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="txt" href="#E12:30_1_1"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="#E15:30_1_1"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<a NAME="E5:30_1_1_6"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:30_1_1"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="txt" href="#E1:30_1_1_6"><span class="lab"><font color="Green" title="E54">A47</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E6:30_1_1_6"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E7:30_1_1_6"/><span class="kw">then </span>
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b9</font> or <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_1"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="txt" href="#E10:30_1_1"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="txt" href="#E14:30_1_1"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:30_1_1_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E2:30_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E4:30_1_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E7:30_1_1"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E13:30_1_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E14:30_1_1"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E15:30_1_1"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="txt" href="#E16:30_1_1"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="txt" href="#E17:30_1_1"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="txt" href="#E20:30_1_1"><span class="lab"><font color="Green" title="E35">A22</font></span></a>, <a class="txt" href="#E4:30_1_1_6"><span class="lab"><font color="Green" title="E55">A48</font></span></a>, <a class="ref" href="aff_1.html#T11" title="AFF_1:th.11">AFF_1:11</a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E45:30_1_1"/><span class="lab"><font color="Green" title="E54">A49</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_1_1_7"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:30_1_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c9">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:30_1_1"><span class="lab"><font color="Green" title="E52">A45</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E3:30_1_1_7"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c9">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E4:30_1_1_7"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E5:30_1_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c11">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="#E7:30_1_1"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E8:30_1_1"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="txt" href="#E13:30_1_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E44:30_1_1"><span class="lab"><font color="Green" title="E53">A46</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="E6:30_1_1_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E12:30_1_1"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="#E13:30_1_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E15:30_1_1"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="txt" href="#E17:30_1_1"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="txt" href="#E23:30_1_1"><span class="lab"><font color="Green" title="E38">A28</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E46:30_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A50</font></span>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c16">x</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c11">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c16">x</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="E47:30_1_1"/><span class="lab"><font color="Green" title="E56">A51</font></span>: 
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c16">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:30_1_1"><span class="lab"><font color="Green" title="E44">A37</font></span></a>, <a class="txt" href="#E32:30_1_1"><span class="lab"><font color="Green" title="E45">A38</font></span></a>, <a class="txt" href="#E33:30_1_1"><span class="lab"><font color="Green" title="E46">A39</font></span></a>, <a class="txt" href="#E40:30_1_1"><span class="lab"><font color="Green" title="E51">A44</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/>
<a NAME="E48:30_1_1"/><span class="lab"><font color="Green" title="E57">A52</font></span>: 
<font color="Maroon" title="c16">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="c11">c9</font>,<font color="Maroon" title="c16">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:30_1_1"><span class="lab"><font color="Green" title="E54">A49</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E49:30_1_1"/><span class="lab"><font color="Green" title="E58">A53</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_1_1_8"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</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="E2:30_1_1_8"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:30_1_1"><span class="lab"><font color="Green" title="E44">A37</font></span></a>, <a class="txt" href="#E32:30_1_1"><span class="lab"><font color="Green" title="E45">A38</font></span></a>, <a class="txt" href="#E33:30_1_1"><span class="lab"><font color="Green" title="E46">A39</font></span></a>, <a class="txt" href="#E40:30_1_1"><span class="lab"><font color="Green" title="E51">A44</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/>
<a NAME="E3:30_1_1_8"/><span class="kw">then </span>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:30_1_1"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="txt" href="#E26:30_1_1"><span class="lab"><font color="Green" title="E41">A32</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E4:30_1_1_8"/><span class="kw">then </span>
<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c9">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E5:30_1_1_8"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:30_1_1_8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:30_1_1"><span class="lab"><font color="Green" title="E42">A33</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</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="E50:30_1_1"/>
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c16">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_1_1_9"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c16">x</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="E2:30_1_1_9"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c16">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="E3:30_1_1_9"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:30_1_1"><span class="lab"><font color="Green" title="E52">A45</font></span></a>, <a class="txt" href="#E49:30_1_1"><span class="lab"><font color="Green" title="E58">A53</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E4:30_1_1_9"/><span class="kw">then </span>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:30_1_1"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="txt" href="#E26:30_1_1"><span class="lab"><font color="Green" title="E41">A32</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:30_1_1_9"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:30_1_1"><span class="lab"><font color="Green" title="E42">A33</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E51:30_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">o</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="c11">c9</font>,<font color="Maroon" title="c16">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E2:30_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E4:30_1_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E5:30_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E7:30_1_1"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E8:30_1_1"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="txt" href="#E9:30_1_1"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="txt" href="#E10:30_1_1"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="txt" href="#E13:30_1_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E14:30_1_1"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E16:30_1_1"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="txt" href="#E18:30_1_1"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="txt" href="#E43:30_1_1"><span class="lab"><font color="Green" title="E52">A45</font></span></a>, <a class="txt" href="#E47:30_1_1"><span class="lab"><font color="Green" title="E56">A51</font></span></a>, <a class="txt" href="#E45:30_1_1"><span class="lab"><font color="Green" title="E54">A49</font></span></a>, <a class="txt" href="#E46:30_1_1"><span class="lab"><font color="Green" title="E55">A50</font></span></a>, <a class="txt" href="#E48:30_1_1"><span class="lab"><font color="Green" title="E57">A52</font></span></a>, <a class="ref" href="aff_2.html#D5" target="_self" title="AFF_2:def.5">Def5</a></span>;<br/>
<a NAME="E52:30_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">x</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="#E3:30_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E12:30_1_1"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="#E15:30_1_1"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="txt" href="#E23:30_1_1"><span class="lab"><font color="Green" title="E38">A28</font></span></a>, <a class="txt" href="#E46:30_1_1"><span class="lab"><font color="Green" title="E55">A50</font></span></a>, <a class="txt" href="#E48:30_1_1"><span class="lab"><font color="Green" title="E57">A52</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E53:30_1_1"/><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="c6">a</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_1"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="txt" href="#E15:30_1_1"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="txt" href="#E20:30_1_1"><span class="lab"><font color="Green" title="E35">A22</font></span></a>, <a class="txt" href="#E36:30_1_1"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E25:30_1_1"><span class="lab"><font color="Green" title="E40">A31</font></span></a>, <a class="txt" href="#E39:30_1_1"><span class="lab"><font color="Green" title="E50">A43</font></span></a>, <a class="txt" href="#E47:30_1_1"><span class="lab"><font color="Green" title="E56">A51</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="E54:30_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E4:30_1_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E7:30_1_1"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E13:30_1_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E15:30_1_1"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="txt" href="#E17:30_1_1"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="txt" href="#E34:30_1_1"><span class="lab"><font color="Green" title="E47">A40</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:30_2"/><span class="lab"><font color="Green" title="E15">A54</font></span>: 
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V5" title="AFF_2:attr.5">satisfying_DES_1</a> </span><br/><span class="kw">thus </span><a NAME="E2:30_2"/>
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V5" title="AFF_2:attr.5">satisfying_DES_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="30_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="aff_2.html#D5" target="_self" title="AFF_2:def.5">AFF_2:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><br/> for <font color="Olive" title="b3">o</font>, <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font>, <font color="Olive" title="b6">c</font>, <font color="Olive" title="b7">a9</font>, <font color="Olive" title="b8">b9</font>, <font color="Olive" title="b9">c9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>  st <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">a</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">b</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">c</font> &amp; <font color="Olive" title="b4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b7">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b8">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b9">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp; <font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b4">a</font>,<font color="Olive" title="b5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">a9</font>,<font color="Olive" title="b8">b9</font> &amp; <font color="Olive" title="b4">a</font>,<font color="Olive" title="b6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">a9</font>,<font color="Olive" title="b9">c9</font> &amp; <font color="Olive" title="b5">b</font>,<font color="Olive" title="b6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b8">b9</font>,<font color="Olive" title="b9">c9</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b4">a</font>,<font color="Olive" title="b5">b</font>,<font color="Olive" title="b6">c</font> &amp; <font color="Olive" title="b6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b9">c9</font> holds <br/><font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font>, <font color="Olive" title="b5">a9</font>, <font color="Olive" title="b6">b9</font>, <font color="Olive" title="b7">c9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>  st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">c</font> &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b5">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b6">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b7">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">a9</font>,<font color="Olive" title="b6">b9</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">a9</font>,<font color="Olive" title="b7">c9</font> &amp; <font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">b9</font>,<font color="Olive" title="b7">c9</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font> &amp; <font color="Olive" title="b4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">c9</font> holds <br/><font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">o</font>, <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c9">a9</font>, <font color="Maroon" title="c10">b9</font>, <font color="Maroon" title="c11">c9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c10">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c11">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> &amp; <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font> &amp; <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c8">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c9</font> implies <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )</span><br/>



















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:30_2_1"/><span class="lab"><font color="Green" title="E16">A55</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E2:30_2_1"/><span class="lab"><font color="Green" title="E17">A56</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E3:30_2_1"/><span class="lab"><font color="Green" title="E18">A57</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">and </span><br/><a NAME="E4:30_2_1"/><span class="lab"><font color="Green" title="E19">A58</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">and </span><br/><a NAME="E5:30_2_1"/>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c</font>
 <span class="kw">and </span><br/><a NAME="E6:30_2_1"/><span class="lab"><font color="Green" title="E20">A59</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E7:30_2_1"/><span class="lab"><font color="Green" title="E21">A60</font></span>: 
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E8:30_2_1"/><span class="lab"><font color="Green" title="E22">A61</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E9:30_2_1"/><span class="lab"><font color="Green" title="E23">A62</font></span>: 
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E10:30_2_1"/><span class="lab"><font color="Green" title="E24">A63</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E11:30_2_1"/><span class="lab"><font color="Green" title="E25">A64</font></span>: 
<font color="Maroon" title="c11">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E12:30_2_1"/><span class="lab"><font color="Green" title="E26">A65</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="E13:30_2_1"/><span class="lab"><font color="Green" title="E27">A66</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="E14:30_2_1"/><span class="lab"><font color="Green" title="E28">A67</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="E15:30_2_1"/><span class="lab"><font color="Green" title="E29">A68</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="E16:30_2_1"/><span class="lab"><font color="Green" title="E30">A69</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="E17:30_2_1"/><span class="lab"><font color="Green" title="E31">A70</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font>
 <span class="kw">and </span><br/><a NAME="E18:30_2_1"/><span class="lab"><font color="Green" title="E32">A71</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>
 <span class="kw">and </span><br/><a NAME="E19:30_2_1"/><span class="lab"><font color="Green" title="E33">A72</font></span>: 
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 <span class="kw">and </span><br/><a NAME="E20:30_2_1"/><span class="lab"><font color="Green" title="E34">A73</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>
 <span class="kw">and </span><br/><a NAME="E21:30_2_1"/><span class="lab"><font color="Green" title="E35">A74</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c12">K</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c11">c9</font>;<br/>
<span class="kw">assume </span><a NAME="E22:30_2_1"/><span class="lab"><font color="Green" title="E36">A75</font></span>: 
not <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 ; <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="E23:30_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A76</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c11">c9</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_2_1"><span class="lab"><font color="Green" title="E25">A64</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E24:30_2_1"/><span class="lab"><font color="Green" title="E38">A77</font></span>: 
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_2_1_1"/><span class="lab"><font color="Green" title="E39">A78</font></span>: 
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:30_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:30_2_1"><span class="lab"><font color="Green" title="E33">A72</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E3:30_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> or <font color="Maroon" title="c9">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:30_2_1"><span class="lab"><font color="Green" title="E31">A70</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E4:30_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> or <font color="Maroon" title="c9">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E5:30_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">b</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> or <font color="Maroon" title="c9">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:30_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_2_1"><span class="lab"><font color="Green" title="E16">A55</font></span></a>, <a class="txt" href="#E2:30_2_1"><span class="lab"><font color="Green" title="E17">A56</font></span></a>, <a class="txt" href="#E7:30_2_1"><span class="lab"><font color="Green" title="E21">A60</font></span></a>, <a class="txt" href="#E9:30_2_1"><span class="lab"><font color="Green" title="E23">A62</font></span></a>, <a class="txt" href="#E11:30_2_1"><span class="lab"><font color="Green" title="E25">A64</font></span></a>, <a class="txt" href="#E12:30_2_1"><span class="lab"><font color="Green" title="E26">A65</font></span></a>, <a class="txt" href="#E13:30_2_1"><span class="lab"><font color="Green" title="E27">A66</font></span></a>, <a class="txt" href="#E15:30_2_1"><span class="lab"><font color="Green" title="E29">A68</font></span></a>, <a class="txt" href="#E20:30_2_1"><span class="lab"><font color="Green" title="E34">A73</font></span></a>, <a class="txt" href="#E22:30_2_1"><span class="lab"><font color="Green" title="E36">A75</font></span></a>, <a class="txt" href="#E1:30_2_1_1"><span class="lab"><font color="Green" title="E39">A78</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E25:30_2_1"/><span class="lab"><font color="Green" title="E39">A79</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="c5">o</font>,<font color="Maroon" title="c11">c9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_2_1_2"/>
<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="c5">o</font>,<font color="Maroon" title="c11">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:30_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A80</font></span>: 
<font color="Maroon" title="c11">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="#E1:30_2_1"><span class="lab"><font color="Green" title="E16">A55</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E3:30_2_1_2"/>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:30_2_1"><span class="lab"><font color="Green" title="E32">A71</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E4:30_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">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="#E6:30_2_1"><span class="lab"><font color="Green" title="E20">A59</font></span></a>, <a class="txt" href="#E7:30_2_1"><span class="lab"><font color="Green" title="E21">A60</font></span></a>, <a class="txt" href="#E12:30_2_1"><span class="lab"><font color="Green" title="E26">A65</font></span></a>, <a class="txt" href="#E24:30_2_1"><span class="lab"><font color="Green" title="E38">A77</font></span></a>, <a class="txt" href="#E2:30_2_1_2"><span class="lab"><font color="Green" title="E40">A80</font></span></a>, <a class="ref" href="aff_1.html#T62" title="AFF_1:th.62">AFF_1:62</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:30_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_2_1"><span class="lab"><font color="Green" title="E24">A63</font></span></a>, <a class="txt" href="#E11:30_2_1"><span class="lab"><font color="Green" title="E25">A64</font></span></a>, <a class="txt" href="#E12:30_2_1"><span class="lab"><font color="Green" title="E26">A65</font></span></a>, <a class="txt" href="#E14:30_2_1"><span class="lab"><font color="Green" title="E28">A67</font></span></a>, <a class="txt" href="#E16:30_2_1"><span class="lab"><font color="Green" title="E30">A69</font></span></a>, <a class="txt" href="#E21:30_2_1"><span class="lab"><font color="Green" title="E35">A74</font></span></a>, <a class="txt" href="#E2:30_2_1_2"><span class="lab"><font color="Green" title="E40">A80</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E26:30_2_1"/><span class="lab"><font color="Green" title="E40">A81</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:30_2_1"><span class="lab"><font color="Green" title="E34">A73</font></span></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E27:30_2_1"/><span class="lab"><font color="Green" title="E41">A82</font></span>: 
<font color="Maroon" title="c5">o</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">o</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_2_1"><span class="lab"><font color="Green" title="E25">A64</font></span></a>, <a class="txt" href="#E22:30_2_1"><span class="lab"><font color="Green" title="E36">A75</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E28:30_2_1"/><span class="lab"><font color="Green" title="E42">A83</font></span>: 
<font color="Maroon" title="c11">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_2_1"><span class="lab"><font color="Green" title="E25">A64</font></span></a>, <a class="txt" href="#E22:30_2_1"><span class="lab"><font color="Green" title="E36">A75</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E29:30_2_1"/>
not <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</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="c5">o</font>,<font color="Maroon" title="c11">c9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_2_1_3"/>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</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="c5">o</font>,<font color="Maroon" title="c11">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:30_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</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="c5">o</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:30_2_1"><span class="lab"><font color="Green" title="E32">A71</font></span></a>, <a class="txt" href="#E26:30_2_1"><span class="lab"><font color="Green" title="E40">A81</font></span></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a></span>;<br/>
<a NAME="E3:30_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c9">a9</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="c5">o</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T48" title="AFF_1:th.48">AFF_1:48</a></span>;<br/>
<a NAME="E4:30_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:30_2_1"><span class="lab"><font color="Green" title="E37">A76</font></span></a>, <a class="txt" href="#E28:30_2_1"><span class="lab"><font color="Green" title="E42">A83</font></span></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></span>;<br/>
<a NAME="E5:30_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A84</font></span>: 
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_2_1"><span class="lab"><font color="Green" title="E16">A55</font></span></a>, <a class="txt" href="#E2:30_2_1"><span class="lab"><font color="Green" title="E17">A56</font></span></a>, <a class="txt" href="#E7:30_2_1"><span class="lab"><font color="Green" title="E21">A60</font></span></a>, <a class="txt" href="#E12:30_2_1"><span class="lab"><font color="Green" title="E26">A65</font></span></a>, <a class="txt" href="#E23:30_2_1"><span class="lab"><font color="Green" title="E37">A76</font></span></a>, <a class="txt" href="#E27:30_2_1"><span class="lab"><font color="Green" title="E41">A82</font></span></a>, <a class="txt" href="#E25:30_2_1"><span class="lab"><font color="Green" title="E39">A79</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E6:30_2_1_3"/>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:30_2_1"><span class="lab"><font color="Green" title="E31">A70</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E7:30_2_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c9">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b9</font> or <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_2_1"><span class="lab"><font color="Green" title="E22">A61</font></span></a>, <a class="txt" href="#E9:30_2_1"><span class="lab"><font color="Green" title="E23">A62</font></span></a>, <a class="txt" href="#E13:30_2_1"><span class="lab"><font color="Green" title="E27">A66</font></span></a>, <a class="txt" href="#E5:30_2_1_3"><span class="lab"><font color="Green" title="E43">A84</font></span></a>, <a class="ref" href="aff_1.html#T62" title="AFF_1:th.62">AFF_1:62</a></span>;<br/>
<a NAME="E8:30_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_2_1"><span class="lab"><font color="Green" title="E16">A55</font></span></a>, <a class="txt" href="#E2:30_2_1"><span class="lab"><font color="Green" title="E17">A56</font></span></a>, <a class="txt" href="#E3:30_2_1"><span class="lab"><font color="Green" title="E18">A57</font></span></a>, <a class="txt" href="#E6:30_2_1"><span class="lab"><font color="Green" title="E20">A59</font></span></a>, <a class="txt" href="#E12:30_2_1"><span class="lab"><font color="Green" title="E26">A65</font></span></a>, <a class="txt" href="#E13:30_2_1"><span class="lab"><font color="Green" title="E27">A66</font></span></a>, <a class="txt" href="#E15:30_2_1"><span class="lab"><font color="Green" title="E29">A68</font></span></a>, <a class="txt" href="#E18:30_2_1"><span class="lab"><font color="Green" title="E32">A71</font></span></a>, <a class="txt" href="#E19:30_2_1"><span class="lab"><font color="Green" title="E33">A72</font></span></a>, <a class="txt" href="#E24:30_2_1"><span class="lab"><font color="Green" title="E38">A77</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E9:30_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E10:30_2_1_3"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:30_2_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:30_2_1"><span class="lab"><font color="Green" title="E34">A73</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">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="E31:30_2_1"/><span class="lab"><font color="Green" title="E43">A85</font></span>: 
<font color="Maroon" title="c13">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">o</font>,<font color="Maroon" title="c11">c9</font>
 <span class="kw">and </span><br/><a NAME="E32:30_2_1"/><span class="lab"><font color="Green" title="E44">A86</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:30_2_1"><span class="lab"><font color="Green" title="E37">A76</font></span></a>, <a class="ref" href="aff_1.html#T73" title="AFF_1:th.73">AFF_1:73</a></span>;<br/>
<a NAME="E33:30_2_1"/><span class="lab"><font color="Green" title="E45">A87</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_2_1_4"/>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</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="E2:30_2_1_4"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:30_2_1"><span class="lab"><font color="Green" title="E44">A86</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E3:30_2_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A88</font></span>: 
<font color="Maroon" title="c8">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="#E1:30_2_1"><span class="lab"><font color="Green" title="E16">A55</font></span></a>, <a class="txt" href="#E3:30_2_1"><span class="lab"><font color="Green" title="E18">A57</font></span></a>, <a class="txt" href="#E6:30_2_1"><span class="lab"><font color="Green" title="E20">A59</font></span></a>, <a class="txt" href="#E12:30_2_1"><span class="lab"><font color="Green" title="E26">A65</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<a NAME="E4:30_2_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c11">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="#E6:30_2_1"><span class="lab"><font color="Green" title="E20">A59</font></span></a>, <a class="txt" href="#E7:30_2_1"><span class="lab"><font color="Green" title="E21">A60</font></span></a>, <a class="txt" href="#E12:30_2_1"><span class="lab"><font color="Green" title="E26">A65</font></span></a>, <a class="txt" href="#E18:30_2_1"><span class="lab"><font color="Green" title="E32">A71</font></span></a>, <a class="txt" href="#E26:30_2_1"><span class="lab"><font color="Green" title="E40">A81</font></span></a>, <a class="ref" href="aff_1.html#T62" title="AFF_1:th.62">AFF_1:62</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:30_2_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_2_1"><span class="lab"><font color="Green" title="E24">A63</font></span></a>, <a class="txt" href="#E11:30_2_1"><span class="lab"><font color="Green" title="E25">A64</font></span></a>, <a class="txt" href="#E12:30_2_1"><span class="lab"><font color="Green" title="E26">A65</font></span></a>, <a class="txt" href="#E14:30_2_1"><span class="lab"><font color="Green" title="E28">A67</font></span></a>, <a class="txt" href="#E16:30_2_1"><span class="lab"><font color="Green" title="E30">A69</font></span></a>, <a class="txt" href="#E21:30_2_1"><span class="lab"><font color="Green" title="E35">A74</font></span></a>, <a class="txt" href="#E3:30_2_1_4"><span class="lab"><font color="Green" title="E46">A88</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E34:30_2_1"/><span class="lab"><font color="Green" title="E46">A89</font></span>: 
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_2_1_5"/>
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:30_2_1_5"/><span class="kw">then </span>
( <font color="Maroon" title="c9">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b9</font> or <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:30_2_1"><span class="lab"><font color="Green" title="E31">A70</font></span></a>, <a class="txt" href="#E18:30_2_1"><span class="lab"><font color="Green" title="E32">A71</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E3:30_2_1_5"/><span class="kw">then </span>
( <font color="Maroon" title="c9">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b9</font> or  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c7">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E4:30_2_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:30_2_1"><span class="lab"><font color="Green" title="E32">A71</font></span></a>, <a class="txt" href="#E19:30_2_1"><span class="lab"><font color="Green" title="E33">A72</font></span></a>, <a class="txt" href="#E20:30_2_1"><span class="lab"><font color="Green" title="E34">A73</font></span></a>, <a class="txt" href="#E24:30_2_1"><span class="lab"><font color="Green" title="E38">A77</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E5:30_2_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E6:30_2_1_5"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:30_2_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:30_2_1"><span class="lab"><font color="Green" title="E34">A73</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</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="E35:30_2_1"/><span class="lab"><font color="Green" title="E47">A90</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c13">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:30_2_1"><span class="lab"><font color="Green" title="E44">A86</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="E36:30_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c13">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:30_2_1"><span class="lab"><font color="Green" title="E32">A71</font></span></a>, <a class="txt" href="#E26:30_2_1"><span class="lab"><font color="Green" title="E40">A81</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E37:30_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c13">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_2"><span class="lab"><font color="Green" title="E15">A54</font></span></a>, <a class="txt" href="#E1:30_2_1"><span class="lab"><font color="Green" title="E16">A55</font></span></a>, <a class="txt" href="#E2:30_2_1"><span class="lab"><font color="Green" title="E17">A56</font></span></a>, <a class="txt" href="#E3:30_2_1"><span class="lab"><font color="Green" title="E18">A57</font></span></a>, <a class="txt" href="#E4:30_2_1"><span class="lab"><font color="Green" title="E19">A58</font></span></a>, <a class="txt" href="#E6:30_2_1"><span class="lab"><font color="Green" title="E20">A59</font></span></a>, <a class="txt" href="#E7:30_2_1"><span class="lab"><font color="Green" title="E21">A60</font></span></a>, <a class="txt" href="#E8:30_2_1"><span class="lab"><font color="Green" title="E22">A61</font></span></a>, <a class="txt" href="#E9:30_2_1"><span class="lab"><font color="Green" title="E23">A62</font></span></a>, <a class="txt" href="#E12:30_2_1"><span class="lab"><font color="Green" title="E26">A65</font></span></a>, <a class="txt" href="#E13:30_2_1"><span class="lab"><font color="Green" title="E27">A66</font></span></a>, <a class="txt" href="#E15:30_2_1"><span class="lab"><font color="Green" title="E29">A68</font></span></a>, <a class="txt" href="#E17:30_2_1"><span class="lab"><font color="Green" title="E31">A70</font></span></a>, <a class="txt" href="#E23:30_2_1"><span class="lab"><font color="Green" title="E37">A76</font></span></a>, <a class="txt" href="#E27:30_2_1"><span class="lab"><font color="Green" title="E41">A82</font></span></a>, <a class="txt" href="#E28:30_2_1"><span class="lab"><font color="Green" title="E42">A83</font></span></a>, <a class="txt" href="#E25:30_2_1"><span class="lab"><font color="Green" title="E39">A79</font></span></a>, <a class="txt" href="#E31:30_2_1"><span class="lab"><font color="Green" title="E43">A85</font></span></a>, <a class="txt" href="#E33:30_2_1"><span class="lab"><font color="Green" title="E45">A87</font></span></a>, <a class="ref" href="aff_2.html#D4" target="_self" title="AFF_2:def.4">Def4</a></span>;<br/>
<a NAME="E38:30_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A91</font></span>: 
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c13">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:30_2_1"><span class="lab"><font color="Green" title="E33">A72</font></span></a>, <a class="txt" href="#E34:30_2_1"><span class="lab"><font color="Green" title="E46">A89</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E39:30_2_1"/><span class="lab"><font color="Green" title="E49">A92</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c13">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_2_1_6"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c13">x</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="E2:30_2_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c13">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="E3:30_2_1_6"/><span class="kw">then </span>
( <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> or <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:30_2_1"><span class="lab"><font color="Green" title="E47">A90</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:30_2_1_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_2_1"><span class="lab"><font color="Green" title="E16">A55</font></span></a>, <a class="txt" href="#E3:30_2_1"><span class="lab"><font color="Green" title="E18">A57</font></span></a>, <a class="txt" href="#E6:30_2_1"><span class="lab"><font color="Green" title="E20">A59</font></span></a>, <a class="txt" href="#E12:30_2_1"><span class="lab"><font color="Green" title="E26">A65</font></span></a>, <a class="txt" href="#E20:30_2_1"><span class="lab"><font color="Green" title="E34">A73</font></span></a>, <a class="txt" href="#E23:30_2_1"><span class="lab"><font color="Green" title="E37">A76</font></span></a>, <a class="txt" href="#E27:30_2_1"><span class="lab"><font color="Green" title="E41">A82</font></span></a>, <a class="txt" href="#E25:30_2_1"><span class="lab"><font color="Green" title="E39">A79</font></span></a>, <a class="txt" href="#E31:30_2_1"><span class="lab"><font color="Green" title="E43">A85</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E40:30_2_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:30_2_1"><span class="lab"><font color="Green" title="E44">A86</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E41:30_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:30_2_1"><span class="lab"><font color="Green" title="E43">A85</font></span></a>, <a class="txt" href="#E39:30_2_1"><span class="lab"><font color="Green" title="E49">A92</font></span></a>, <a class="txt" href="#E38:30_2_1"><span class="lab"><font color="Green" title="E48">A91</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="E42:30_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_2_1"><span class="lab"><font color="Green" title="E24">A63</font></span></a>, <a class="txt" href="#E11:30_2_1"><span class="lab"><font color="Green" title="E25">A64</font></span></a>, <a class="txt" href="#E14:30_2_1"><span class="lab"><font color="Green" title="E28">A67</font></span></a>, <a class="txt" href="#E21:30_2_1"><span class="lab"><font color="Green" title="E35">A74</font></span></a>, <a class="txt" href="#E22:30_2_1"><span class="lab"><font color="Green" title="E36">A75</font></span></a>, <a class="txt" href="#E23:30_2_1"><span class="lab"><font color="Green" title="E37">A76</font></span></a>, <a class="txt" href="#E27:30_2_1"><span class="lab"><font color="Green" title="E41">A82</font></span></a>, <a class="txt" href="#E28:30_2_1"><span class="lab"><font color="Green" title="E42">A83</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:30"/>
( <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a>  iff <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V5" title="AFF_2:attr.5">satisfying_DES_1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E14">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>
