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

<b>let </b><font color="Maroon" title="c1">OAP</font> be   <a href="analoaf.html#NM1" title="ANALOAF:NM.1">OAffinSpace</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">OAP</font> is <a href="papdesaf.html#V6" title="PAPDESAF:attr.6">satisfying_DES_1</a> implies  <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c1">OAP</font> is <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a> )</span><br/>

<b>assume </b><a NAME="E1:15"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c1">OAP</font> is <a href="papdesaf.html#V6" title="PAPDESAF:attr.6">satisfying_DES_1</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c1">OAP</font> is <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a></span><br/>

<a NAME="E2:15"/><b>then </b><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c1">OAP</font> is <a href="papdesaf.html#V5" title="PAPDESAF:attr.5">satisfying_DES</a>
 
<b>by </b><i><a class="ref" href="papdesaf.html#T11" target="_self" title="PAPDESAF:th.11">Th11</a></i>;<br/>
<b>set </b><font color="Maroon" title="c2">AP</font> =  <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c1">OAP</font>;<br/>
<a NAME="E3:15"/>
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">P</font>, <font color="Olive" title="b3">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c1">OAP</font></span>)</span><br/> for <font color="Olive" title="b4">o</font>, <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">b</font>, <font color="Olive" title="b7">c</font>, <font color="Olive" title="b8">a'</font>, <font color="Olive" title="b9">b'</font>, <font color="Olive" title="b10">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c1">OAP</font></span>)</span>  st <font color="Olive" title="b4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">C</font> &amp; <font color="Olive" title="b4">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="b4">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="b4">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="b5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b8">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b9">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">C</font> &amp; <font color="Olive" title="b10">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">C</font> &amp; <font color="Olive" title="b1">A</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Olive" title="b2">P</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Olive" title="b3">C</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">C</font> &amp; <font color="Olive" title="b5">a</font>,<font color="Olive" title="b6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b8">a'</font>,<font color="Olive" title="b9">b'</font> &amp; <font color="Olive" title="b5">a</font>,<font color="Olive" title="b7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b8">a'</font>,<font color="Olive" title="b10">c'</font> holds <br/><font color="Olive" title="b6">b</font>,<font color="Olive" title="b7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b9">b'</font>,<font color="Olive" title="b10">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">P</font>, <font color="Maroon" title="c5">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c1">OAP</font></span>)</span>; <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 <span class="p1">(<span class="default"><a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c1">OAP</font></span>)</span>  st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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="c4">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="c5">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="c3">A</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">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="c4">P</font> &amp; <font color="Olive" title="b6">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">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="c5">C</font> &amp; <font color="Olive" title="b7">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c3">A</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c4">P</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c5">C</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">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">a'</font>,<font color="Olive" title="b6">b'</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">a'</font>,<font color="Olive" title="b7">c'</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">b'</font>,<font color="Olive" title="b7">c'</font></span><br/><b>let </b><font color="Maroon" title="c6">o</font>, <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">c</font>, <font color="Maroon" title="c10">a'</font>, <font color="Maroon" title="c11">b'</font>, <font color="Maroon" title="c12">c'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c1">OAP</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c10">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c11">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c12">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c3">A</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c4">P</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c5">C</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">a'</font>,<font color="Maroon" title="c11">b'</font> &amp; <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">a'</font>,<font color="Maroon" title="c12">c'</font> implies <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b'</font>,<font color="Maroon" title="c12">c'</font> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:15_1"/><i><font color="Green" title="E18">A3</font></i>: 
( <font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> )
 <b>and </b><br/><a NAME="E2:15_1"/><i><font color="Green" title="E19">A4</font></i>: 
( <font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font> )
 <b>and </b><br/><a NAME="E3:15_1"/><i><font color="Green" title="E20">A5</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c10">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c11">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c12">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> )
 <b>and </b><br/><a NAME="E4:15_1"/><i><font color="Green" title="E21">A6</font></i>: 
( <font color="Maroon" title="c3">A</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c4">P</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c5">C</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 <b>and </b><br/><a NAME="E5:15_1"/><i><font color="Green" title="E22">A7</font></i>: 
( <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">C</font> )
 <b>and </b><br/><a NAME="E6:15_1"/><i><font color="Green" title="E23">A8</font></i>: 
( <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">a'</font>,<font color="Maroon" title="c11">b'</font> &amp; <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">a'</font>,<font color="Maroon" title="c12">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">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b'</font>,<font color="Maroon" title="c12">c'</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c13">o1</font> = <font color="Maroon" title="c6">o</font>, <font color="Maroon" title="c14">a1</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c15">b1</font> = <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c16">c1</font> = <font color="Maroon" title="c9">c</font>, <font color="Maroon" title="c17">a1'</font> = <font color="Maroon" title="c10">a'</font>, <font color="Maroon" title="c18">b1'</font> = <font color="Maroon" title="c11">b'</font>, <font color="Maroon" title="c19">c1'</font> = <font color="Maroon" title="c12">c'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAP</font> <b>by </b><i><a class="ref" href="papdesaf.html#T2" target="_self" title="PAPDESAF:th.2">Th2</a></i>;<br/>
<a NAME="E8:15_1"/><i><font color="Green" title="E24">A9</font></i>: 
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c17">a1'</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c18">b1'</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c16">c1</font>,<font color="Maroon" title="c19">c1'</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_1"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c10">a'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c11">b'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c12">c'</font> )
 
<b>by </b><i><a class="txt" href="#E1:15_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="#E3:15_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="#E4:15_1"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<b>hence </b><a NAME="E2:15_1_1"/>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c17">a1'</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c18">b1'</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c16">c1</font>,<font color="Maroon" title="c19">c1'</font> )
 <b>by </b><i><a class="ref" href="papdesaf.html#T3" target="_self" title="PAPDESAF:th.3">Th3</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"/><i><font color="Green" title="E25">A10</font></i>: 
( not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c15">b1</font> &amp; not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c16">c1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_2"/><i><font color="Green" title="E26">A11</font></i>: 
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c15">b1</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c16">c1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<div><i><font color="Green" title="E27">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_2_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c20">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c1">OAP</font></span>)</span><b> such that </b><br/><a NAME="E3:15_1_2_1"/><i><font color="Green" title="E28">A13</font></i>: 
( <font color="Maroon" title="c20">X</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">X</font> &amp; <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">X</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">X</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/><a NAME="E4:15_1_2_1"/>
( <font color="Maroon" title="c20">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c20">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> )
 <b>by </b><i><a class="txt" href="#E1:15_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="#E2:15_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="#E3:15_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="#E4:15_1"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="#E3:15_1_2_1"><i><font color="Green" title="E28">A13</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="E5:15_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:15_1"><i><font color="Green" title="E22">A7</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><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_2_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c20">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c1">OAP</font></span>)</span><b> such that </b><br/><a NAME="E3:15_1_2_2"/><i><font color="Green" title="E28">A14</font></i>: 
( <font color="Maroon" title="c20">X</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">X</font> &amp; <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">X</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">X</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/><a NAME="E4:15_1_2_2"/>
( <font color="Maroon" title="c20">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c20">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> )
 <b>by </b><i><a class="txt" href="#E1:15_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="#E2:15_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="#E3:15_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="#E4:15_1"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="#E3:15_1_2_2"><i><font color="Green" title="E28">A14</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="E5:15_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:15_1"><i><font color="Green" title="E22">A7</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><b>end;</b></div>
<b>hence </b><a NAME="E4:15_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:15_1_2"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="#E2:15_1_2"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="papdesaf.html#T3" target="_self" title="PAPDESAF:th.3">Th3</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"/><i><font color="Green" title="E26">A15</font></i>: 
( <font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c13">o1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c17">a1'</font> or <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c14">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c17">a1'</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_3"/>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c17">a1'</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c17">a1'</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c17">a1'</font>,<font color="Maroon" title="c14">a1</font> )
 
<b>by </b><i><a class="txt" href="#E8:15_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="diraf.html#T35" title="DIRAF:th.35">DIRAF:35</a></i>;<br/>
<b>hence </b><a NAME="E2:15_1_3"/>
( <font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c13">o1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c17">a1'</font> or <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c14">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c17">a1'</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</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="E11:15_1"/><i><font color="Green" title="E27">A16</font></i>: 
( <font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c15">b1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c17">a1'</font>,<font color="Maroon" title="c18">b1'</font> &amp; <font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c16">c1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c17">a1'</font>,<font color="Maroon" title="c19">c1'</font> )
 
<b>by </b><i><a class="txt" href="#E6:15_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="diraf.html#T45" title="DIRAF:th.45">DIRAF:45</a></i>;<br/>
<div><i><font color="Green" title="E28">A17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_4"/><i><font color="Green" title="E29">A18</font></i>: 
<font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c13">o1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c17">a1'</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">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b'</font>,<font color="Maroon" title="c12">c'</font></span><br/><a NAME="E2:15_1_4"/><b>then </b>
( <font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c13">o1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c18">b1'</font> &amp; <font color="Maroon" title="c16">c1</font>,<font color="Maroon" title="c13">o1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c19">c1'</font> &amp; <font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c15">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">b1'</font>,<font color="Maroon" title="c17">a1'</font> &amp; <font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c16">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c19">c1'</font>,<font color="Maroon" title="c17">a1'</font> )
 <b>by </b><i><a class="txt" href="#E8:15_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="#E9:15_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="#E11:15_1"><i><font color="Green" title="E27">A16</font></i></a>, <a class="ref" href="papdesaf.html#T12" target="_self" title="PAPDESAF:th.12">Th12</a></i>;<br/><a NAME="E3:15_1_4"/><b>then </b>
<font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c16">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c19">c1'</font>,<font color="Maroon" title="c18">b1'</font>
 <b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="#E9:15_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="#E1:15_1_4"><i><font color="Green" title="E29">A18</font></i></a>, <a class="ref" href="papdesaf.html#D16" target="_self" title="PAPDESAF:def.16">Def16</a></i>;<br/><a NAME="E4:15_1_4"/><b>then </b>
<font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c16">c1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c18">b1'</font>,<font color="Maroon" title="c19">c1'</font>
 <b>by </b><i><a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></i>;<br/><b>hence </b><a NAME="E5:15_1_4"/>
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b'</font>,<font color="Maroon" title="c12">c'</font>
 <b>by </b><i><a class="ref" href="diraf.html#T45" title="DIRAF:th.45">DIRAF:45</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_5"/><i><font color="Green" title="E29">A19</font></i>: 
<font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c14">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c17">a1'</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">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b'</font>,<font color="Maroon" title="c12">c'</font></span><br/><a NAME="E2:15_1_5"/><b>then </b>
( <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c15">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c18">b1'</font> &amp; <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c16">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">o1</font>,<font color="Maroon" title="c19">c1'</font> &amp; <font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c15">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">a1'</font>,<font color="Maroon" title="c18">b1'</font> &amp; <font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c16">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">a1'</font>,<font color="Maroon" title="c19">c1'</font> )
 <b>by </b><i><a class="txt" href="#E8:15_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="#E9:15_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="#E11:15_1"><i><font color="Green" title="E27">A16</font></i></a>, <a class="ref" href="papdesaf.html#T13" target="_self" title="PAPDESAF:th.13">Th13</a></i>;<br/><a NAME="E3:15_1_5"/><b>then </b>
<font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c16">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">b1'</font>,<font color="Maroon" title="c19">c1'</font>
 <b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="#E9:15_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="#E1:15_1_5"><i><font color="Green" title="E29">A19</font></i></a>, <a class="ref" href="papdesaf.html#D15" target="_self" title="PAPDESAF:def.15">Def15</a></i>;<br/><a NAME="E4:15_1_5"/><b>then </b>
<font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c16">c1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c18">b1'</font>,<font color="Maroon" title="c19">c1'</font>
 <b>by </b><i><a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></i>;<br/><b>hence </b><a NAME="E5:15_1_5"/>
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b'</font>,<font color="Maroon" title="c12">c'</font>
 <b>by </b><i><a class="ref" href="diraf.html#T45" title="DIRAF:th.45">DIRAF:45</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>hence </b><a NAME="E14:15_1"/>
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b'</font>,<font color="Maroon" title="c12">c'</font>
 <b>by </b><i><a class="txt" href="#E10:15_1"><i><font color="Green" title="E26">A15</font></i></a>, <a class="txt" href="#E12:15_1"><i><font color="Green" title="E28">A17</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><b>end;</b></div>
<b>hence </b><a NAME="E4:15"/>
 <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c1">OAP</font> is <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a>
 <b>by </b><i><a class="ref" href="aff_2.html#D4" title="AFF_2:def.4">AFF_2:def 4</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>
