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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">AP</font> is <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a> iff <font color="Maroon" title="c1">AP</font> is <a href="aff_2.html#V3" title="AFF_2:attr.3">satisfying_PAP_1</a> )</span><br/>

<div><i><font color="Green" title="E14">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c1">AP</font> is <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">AP</font> is <a href="aff_2.html#V3" title="AFF_2:attr.3">satisfying_PAP_1</a></span><br/><b>thus </b><a NAME="E2:15_1"/>
<font color="Maroon" title="c1">AP</font> is <a href="aff_2.html#V3" title="AFF_2:attr.3">satisfying_PAP_1</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">M</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="aff_2.html#D3" target="_self" title="AFF_2:def.3">AFF_2:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">N</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><br/> for <font color="Olive" title="b2">o</font>, <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">a'</font>, <font color="Olive" title="b7">b'</font>, <font color="Olive" title="b8">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>  st <font color="Maroon" title="c2">M</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Olive" title="b1">N</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">a</font> &amp; <font color="Olive" title="b2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">a'</font> &amp; <font color="Olive" title="b2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">b</font> &amp; <font color="Olive" title="b2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">b'</font> &amp; <font color="Olive" title="b2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">c</font> &amp; <font color="Olive" title="b2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b8">c'</font> &amp; <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b7">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b8">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b7">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">b</font>,<font color="Olive" title="b6">a'</font> &amp; <font color="Olive" title="b4">b</font>,<font color="Olive" title="b8">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">c</font>,<font color="Olive" title="b7">b'</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b8">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">c</font>,<font color="Olive" title="b6">a'</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">c</font> holds <br/><font color="Olive" title="b6">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font></span><br/><b>let </b><font color="Maroon" title="c3">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">a'</font>, <font color="Olive" title="b6">b'</font>, <font color="Olive" title="b7">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>  st <font color="Maroon" title="c2">M</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c3">N</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</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">N</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="b5">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="b6">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="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">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">M</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b6">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b7">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b6">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b5">a'</font> &amp; <font color="Olive" title="b3">b</font>,<font color="Olive" title="b7">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">c</font>,<font color="Olive" title="b6">b'</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b7">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">c</font>,<font color="Olive" title="b5">a'</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">c</font> holds <br/><font color="Olive" title="b5">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font></span><br/><b>let </b><font color="Maroon" title="c4">o</font>, <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">a'</font>, <font color="Maroon" title="c9">b'</font>, <font color="Maroon" title="c10">c'</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">M</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c3">N</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a'</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b'</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c'</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c9">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c10">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a'</font> &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c9">b'</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c10">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">a'</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c</font> implies <font color="Maroon" title="c8">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> )</span><br/>

















<b>assume </b><a NAME="E1:15_1_1"/><i><font color="Green" title="E16">A3</font></i>: 
( <font color="Maroon" title="c2">M</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c3">N</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a'</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b'</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c'</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c9">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c10">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a'</font> &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c9">b'</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c10">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">a'</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font></span><br/>

<b>assume </b><a NAME="E2:15_1_1"/><i><font color="Green" title="E17">A4</font></i>: 
not <font color="Maroon" title="c8">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:15_1_1"/><i><font color="Green" title="E18">A5</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b'</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c'</font> )
 
<b>by </b><i><a class="txt" href="#E1:15_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<a NAME="E4:15_1_1"/><i><font color="Green" title="E19">A6</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_1_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:15_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c10">c'</font>
 
<b>by </b><i><a class="txt" href="#E1:15_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E3:15_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="txt" href="#E1:15_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="aff_1.html#T62" title="AFF_1:th.62">AFF_1:62</a></i>;<br/>
<b>hence </b><a NAME="E4:15_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:15_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:15_1_1"/>
not <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_1_2"/>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:15_1_1_2"/><b>then </b>
( <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">b'</font> )
 
<b>by </b><i><a class="txt" href="#E1:15_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E3:15_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">b'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">N</font>
 
<b>by </b><i><a class="txt" href="#E4:15_1_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a></i>;<br/>
<a NAME="E4:15_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c5">a</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">N</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T48" title="AFF_1:th.48">AFF_1:48</a></i>;<br/>
<a NAME="E5:15_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 
<b>by </b><i><a class="txt" href="#E1:15_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/>
<b>hence </b><a NAME="E6:15_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:15_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c11">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><b> such that </b><br/><a NAME="E7:15_1_1"/><i><font color="Green" title="E20">A7</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c11">x</font> )
 <b>by </b><i><a class="txt" href="#E1:15_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="aff_1.html#T73" title="AFF_1:th.73">AFF_1:73</a></i>;<br/>
<a NAME="E8:15_1_1"/><i><font color="Green" title="E21">A8</font></i>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c11">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_1_3"/>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c11">x</font>
 
<b>by </b><i><a class="txt" href="#E7:15_1_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:15_1_1_3"/>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="#E1:15_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="#E4:15_1_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E9:15_1_1"/><i><font color="Green" title="E22">A9</font></i>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_1_4"/>
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:15_1_1_4"/><b>then </b>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c4">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">b'</font>
 
<b>by </b><i><a class="txt" href="#E8:15_1_1"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E3:15_1_1_4"/><b>then </b>
<font color="Maroon" title="c9">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="txt" href="#E1:15_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="aff_1.html#T62" title="AFF_1:th.62">AFF_1:62</a></i>;<br/>
<b>hence </b><a NAME="E4:15_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:15_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E10:15_1_1"/><b>then </b>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c10">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c11">x</font>
 
<b>by </b><i><a class="txt" href="#E1:15_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="#E1:15_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="#E7:15_1_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="#E8:15_1_1"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="aff_2.html#D2" target="_self" title="AFF_2:def.2">Def2</a></i>;<br/>
<a NAME="E11:15_1_1"/><b>then </b>
<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c11">x</font>
 
<b>by </b><i><a class="txt" href="#E1:15_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="#E3:15_1_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E12:15_1_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c11">x</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E13:15_1_1"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c7">c</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c6">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c11">x</font> )
 
<b>by </b><i><a class="txt" href="#E7:15_1_1"><i><font color="Green" title="E20">A7</font></i></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#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E14:15_1_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c11">x</font>
 
<b>by </b><i><a class="txt" href="#E2:15_1_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="#E7:15_1_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/>
<a NAME="E15:15_1_1"/><b>then </b>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="txt" href="#E1:15_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E16:15_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:15_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="#E7:15_1_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="#E9:15_1_1"><i><font color="Green" title="E22">A9</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_2"/><i><font color="Green" title="E15">A10</font></i>: 
<font color="Maroon" title="c1">AP</font> is <a href="aff_2.html#V3" title="AFF_2:attr.3">satisfying_PAP_1</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">AP</font> is <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a></span><br/><b>thus </b><a NAME="E2:15_2"/>
<font color="Maroon" title="c1">AP</font> is <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">M</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="aff_2.html#D2" target="_self" title="AFF_2:def.2">AFF_2:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">N</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><br/> for <font color="Olive" title="b2">o</font>, <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">a'</font>, <font color="Olive" title="b7">b'</font>, <font color="Olive" title="b8">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>  st <font color="Maroon" title="c2">M</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Olive" title="b1">N</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">a</font> &amp; <font color="Olive" title="b2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">a'</font> &amp; <font color="Olive" title="b2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">b</font> &amp; <font color="Olive" title="b2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">b'</font> &amp; <font color="Olive" title="b2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">c</font> &amp; <font color="Olive" title="b2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b8">c'</font> &amp; <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b6">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b7">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b8">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b7">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">b</font>,<font color="Olive" title="b6">a'</font> &amp; <font color="Olive" title="b4">b</font>,<font color="Olive" title="b8">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">c</font>,<font color="Olive" title="b7">b'</font> holds <br/><font color="Olive" title="b3">a</font>,<font color="Olive" title="b8">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">c</font>,<font color="Olive" title="b6">a'</font></span><br/><b>let </b><font color="Maroon" title="c3">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">a'</font>, <font color="Olive" title="b6">b'</font>, <font color="Olive" title="b7">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>  st <font color="Maroon" title="c2">M</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c3">N</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</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">N</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="b5">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="b6">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="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">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">M</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b5">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b6">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b7">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b6">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b5">a'</font> &amp; <font color="Olive" title="b3">b</font>,<font color="Olive" title="b7">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">c</font>,<font color="Olive" title="b6">b'</font> holds <br/><font color="Olive" title="b2">a</font>,<font color="Olive" title="b7">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">c</font>,<font color="Olive" title="b5">a'</font></span><br/><b>let </b><font color="Maroon" title="c4">o</font>, <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">a'</font>, <font color="Maroon" title="c9">b'</font>, <font color="Maroon" title="c10">c'</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">M</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c3">N</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a'</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b'</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c'</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c8">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c9">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c10">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a'</font> &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c9">b'</font> implies <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c10">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">a'</font> )</span><br/>

















<b>assume </b><a NAME="E1:15_2_1"/><i><font color="Green" title="E16">A11</font></i>: 
( <font color="Maroon" title="c2">M</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c3">N</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a'</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b'</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c'</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c8">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c9">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c10">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a'</font> &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c9">b'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c10">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">a'</font></span><br/>

<b>assume </b><a NAME="E2:15_2_1"/><i><font color="Green" title="E17">A12</font></i>: 
not <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c10">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">a'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:15_2_1"/><b>then </b><i><font color="Green" title="E18">A13</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c'</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a'</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T12" title="AFF_1:th.12">AFF_1:12</a></i>;<br/>
<a NAME="E4:15_2_1"/><i><font color="Green" title="E19">A14</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b'</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a'</font> )
 
<b>by </b><i><a class="txt" href="#E1:15_2_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<a NAME="E5:15_2_1"/><i><font color="Green" title="E20">A15</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_2_1_1"/><i><font color="Green" title="E21">A16</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:15_2_1_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">c'</font>,<font color="Maroon" title="c9">b'</font>
 
<b>by </b><i><a class="txt" href="#E1:15_2_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:15_2_1_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c10">c'</font>,<font color="Maroon" title="c6">b</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E4:15_2_1_1"/><b>then </b>
( <font color="Maroon" title="c9">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c'</font> or <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> )
 
<b>by </b><i><a class="txt" href="#E1:15_2_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E5:15_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:15_2_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="txt" href="#E2:15_2_1"><i><font color="Green" title="E17">A12</font></i></a>, <a class="txt" href="#E1:15_2_1_1"><i><font color="Green" title="E21">A16</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:15_2_1"/><i><font color="Green" title="E21">A17</font></i>: 
<font color="Maroon" title="c9">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_2_1_2"/>
<font color="Maroon" title="c9">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:15_2_1_2"/><b>then </b>
<font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c7">c</font>
 
<b>by </b><i><a class="txt" href="#E1:15_2_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E3:15_2_1_2"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E4:15_2_1_2"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c9">b'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E5:15_2_1_2"/><b>then </b>
<font color="Maroon" title="c9">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="txt" href="#E1:15_2_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="txt" href="#E5:15_2_1"><i><font color="Green" title="E20">A15</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E6:15_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:15_2_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c11">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c10">c'</font>;<br/>
<b>set </b><font color="Maroon" title="c12">P</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a'</font>;<br/>
<a NAME="E7:15_2_1"/><i><font color="Green" title="E22">A18</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c10">c'</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a'</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c10">c'</font> &amp; <font color="Maroon" title="c10">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c10">c'</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a'</font> &amp; <font color="Maroon" title="c8">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">b</font>,<font color="Maroon" title="c8">a'</font> )
 
<b>by </b><i><a class="txt" href="#E3:15_2_1"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="#E4:15_2_1"><i><font color="Green" title="E19">A14</font></i></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">K</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><b> such that </b><br/><a NAME="E9:15_2_1"/><i><font color="Green" title="E23">A19</font></i>: 
( <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">K</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c10">c'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c13">K</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></i>;<br/>
<a NAME="E10:15_2_1"/><i><font color="Green" title="E24">A20</font></i>: 
( <font color="Maroon" title="c13">K</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c13">K</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="c5">a</font>,<font color="Maroon" title="c10">c'</font> )
 
<b>by </b><i><a class="txt" href="#E9:15_2_1"><i><font color="Green" title="E23">A19</font></i></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/>
<a NAME="E11:15_2_1"/>
not  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c13">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_2_1_3"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c13">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:15_2_1_3"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a'</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="c5">a</font>,<font color="Maroon" title="c10">c'</font>
 
<b>by </b><i><a class="txt" href="#E9:15_2_1"><i><font color="Green" title="E23">A19</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E3:15_2_1_3"/><b>then </b>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c10">c'</font>
 
<b>by </b><i><a class="txt" href="#E7:15_2_1"><i><font color="Green" title="E22">A18</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/>
<a NAME="E4:15_2_1_3"/><b>then </b>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c10">c'</font>
 
<b>by </b><i><a class="txt" href="#E1:15_2_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="txt" href="#E4:15_2_1"><i><font color="Green" title="E19">A14</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E5:15_2_1_3"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c10">c'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E6:15_2_1_3"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c10">c'</font>,<font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E7:15_2_1_3"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 
<b>by </b><i><a class="txt" href="#E1:15_2_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="txt" href="#E6:15_2_1"><i><font color="Green" title="E21">A17</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E8:15_2_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:15_2_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c14">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><b> such that </b><br/><a NAME="E13:15_2_1"/><i><font color="Green" title="E25">A21</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a'</font> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">K</font> )
 <b>by </b><i><a class="txt" href="#E7:15_2_1"><i><font color="Green" title="E22">A18</font></i></a>, <a class="txt" href="#E10:15_2_1"><i><font color="Green" title="E24">A20</font></i></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></i>;<br/>
<a NAME="E14:15_2_1"/><i><font color="Green" title="E26">A22</font></i>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c10">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c14">x</font>
 
<b>by </b><i><a class="txt" href="#E7:15_2_1"><i><font color="Green" title="E22">A18</font></i></a>, <a class="txt" href="#E9:15_2_1"><i><font color="Green" title="E23">A19</font></i></a>, <a class="txt" href="#E13:15_2_1"><i><font color="Green" title="E25">A21</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/>
<a NAME="E15:15_2_1"/>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c14">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:15_2_1_4"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c8">a'</font>
 
<b>by </b><i><a class="txt" href="#E7:15_2_1"><i><font color="Green" title="E22">A18</font></i></a>, <a class="txt" href="#E13:15_2_1"><i><font color="Green" title="E25">A21</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E2:15_2_1_4"/><b>then </b>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c14">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:15_2_1_4"/>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="txt" href="#E1:15_2_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="txt" href="#E4:15_2_1"><i><font color="Green" title="E19">A14</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E16:15_2_1"/><b>then </b>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 
<b>by </b><i><a class="txt" href="#E1:15_2"><i><font color="Green" title="E15">A10</font></i></a>, <a class="txt" href="#E1:15_2_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="txt" href="#E5:15_2_1"><i><font color="Green" title="E20">A15</font></i></a>, <a class="txt" href="#E14:15_2_1"><i><font color="Green" title="E26">A22</font></i></a>, <a class="ref" href="aff_2.html#D3" target="_self" title="AFF_2:def.3">Def3</a></i>;<br/>
<a NAME="E17:15_2_1"/><b>then </b>
<font color="Maroon" title="c3">N</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">b</font>,<font color="Maroon" title="c8">a'</font>
 
<b>by </b><i><a class="txt" href="#E1:15_2_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="txt" href="#E2:15_2_1"><i><font color="Green" title="E17">A12</font></i></a>, <a class="txt" href="#E7:15_2_1"><i><font color="Green" title="E22">A18</font></i></a>, <a class="txt" href="#E13:15_2_1"><i><font color="Green" title="E25">A21</font></i></a>, <a class="txt" href="#E14:15_2_1"><i><font color="Green" title="E26">A22</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E18:15_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:15_2_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="txt" href="#E7:15_2_1"><i><font color="Green" title="E22">A18</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:15"/>
( <font color="Maroon" title="c1">AP</font> is <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a> iff <font color="Maroon" title="c1">AP</font> is <a href="aff_2.html#V3" title="AFF_2:attr.3">satisfying_PAP_1</a> )
 <b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E14">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
