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

<b>let </b><font color="Maroon" title="c1">AS</font> be   <a href="diraf.html#NM1" title="DIRAF:NM.1">AffinSpace</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">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><br/> for <font color="Olive" title="b3">X</font>, <font color="Olive" title="b4">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Olive" title="b3">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  &amp; <font color="Olive" title="b4">Y</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">X</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">X</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">Y</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">Y</font> &amp; <font color="Olive" title="b3">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">Y</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="b2">b</font> holds <br/><font color="Olive" title="b3">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b4">Y</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> </span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</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">X</font>, <font color="Olive" title="b2">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Olive" title="b1">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  &amp; <font color="Olive" title="b2">Y</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Y</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Y</font> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">Y</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> holds <br/><font color="Olive" title="b1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">Y</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> </span><br/><b>let </b><font color="Maroon" title="c4">X</font>, <font color="Maroon" title="c5">Y</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  &amp; <font color="Maroon" title="c5">Y</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> &amp; <font color="Maroon" title="c4">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">Y</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> implies <font color="Maroon" title="c4">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:32"/><i><font color="Green" title="E27">A1</font></i>: 
( <font color="Maroon" title="c4">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  &amp; <font color="Maroon" title="c5">Y</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> )
 <b>and </b><br/><a NAME="E2:32"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">Y</font>
 <b>and </b><br/><a NAME="E3:32"/><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">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="c4">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> </span><br/>

<b>set </b><font color="Maroon" title="c6">Q</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>;<br/>
<b>set </b><font color="Maroon" title="c7">Z</font> = <font color="Maroon" title="c4">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y</font>;<br/>
<a NAME="E4:32"/><i><font color="Green" title="E30">A4</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c2">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="c2">a</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c3">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="c2">a</font>,<font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E3:32"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/>
<a NAME="E5:32"/><i><font color="Green" title="E31">A5</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E1:32"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="aff_4.html#E3:32"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></i>;<br/>
<a NAME="E6:32"/><b>then </b><i><font color="Green" title="E32">A6</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E7:32"/>
<font color="Maroon" title="c4">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1"/>
not <font color="Maroon" title="c4">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">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="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:32_1"/><i><font color="Green" title="E33">A7</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y</font>
 <b>and </b><br/><a NAME="E4:32_1"/><i><font color="Green" title="E34">A8</font></i>: 
not <font color="Maroon" title="c8">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="c2">a</font>,<font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:32_1"/><i><font color="Green" title="E35">A9</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E3:32_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">a'</font> = <font color="Maroon" title="c8">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> <b>by </b><i><a class="txt" href="aff_4.html#E3:32_1"><i><font color="Green" title="E33">A7</font></i></a></i>;<br/>
<a NAME="E7:32_1"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> iff <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> iff <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> )</span><br/>

<div><i><font color="Green" title="E36">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_1_1"/><i><font color="Green" title="E37">A11</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_1_1_1"/><i><font color="Green" title="E38">A12</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font></span><br/><b>reconsider </b><font color="Maroon" title="c11">b'</font> = <font color="Maroon" title="c10">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> <b>by </b><i><a class="txt" href="aff_4.html#E1:32_1_1_1"><i><font color="Green" title="E37">A11</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c12">M</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">b'</font>;<br/><a NAME="E3:32_1_1_1_1"/><i><font color="Green" title="E39">A13</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">b'</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c9">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="c9">a'</font>,<font color="Maroon" title="c11">b'</font> &amp; <font color="Maroon" title="c11">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="c9">a'</font>,<font color="Maroon" title="c11">b'</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:32_1_1_1_1"><i><font color="Green" title="E38">A12</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/><a NAME="E4:32_1_1_1_1"/><i><font color="Green" title="E40">A14</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">b'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:32"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="aff_4.html#E5:32_1"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="aff_4.html#E1:32_1_1_1"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="aff_4.html#E1:32_1_1_1_1"><i><font color="Green" title="E38">A12</font></i></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></i>;<br/><div><i><font color="Green" title="E41">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_1_1_1_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">b'</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="c2">a</font>,<font color="Maroon" title="c3">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="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font></span><br/><a NAME="E2:32_1_1_1_1_1"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">b'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:32"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="aff_4.html#E5:32"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="aff_4.html#E5:32_1"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="aff_4.html#E3:32_1_1_1_1"><i><font color="Green" title="E39">A13</font></i></a>, <a class="ref" href="aff_4.html#T23" target="_self" title="AFF_4:th.23">Th23</a></i>;<br/><b>hence </b><a NAME="E3:32_1_1_1_1_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:32_1_1_1_1"><i><font color="Green" title="E39">A13</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="32_1_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_1_1_1_2"/>
not  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">b'</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="c2">a</font>,<font color="Maroon" title="c3">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="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font></span><br/><b>then consider </b><font color="Maroon" title="c13">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E3:32_1_1_1_1_2"/><i><font color="Green" title="E42">A16</font></i>: 
( <font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">b'</font> &amp; <font color="Maroon" title="c13">q</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="c2">a</font>,<font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:32"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="aff_4.html#E4:32"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="aff_4.html#E5:32"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="aff_4.html#E3:32_1_1_1_1"><i><font color="Green" title="E39">A13</font></i></a>, <a class="txt" href="aff_4.html#E4:32_1_1_1_1"><i><font color="Green" title="E40">A14</font></i></a>, <a class="ref" href="aff_4.html#T22" target="_self" title="AFF_4:th.22">Th22</a></i>;<br/><a NAME="E4:32_1_1_1_1_2"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c13">q</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E4:32_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="aff_4.html#E3:32_1_1_1_1"><i><font color="Green" title="E39">A13</font></i></a>, <a class="txt" href="aff_4.html#E3:32_1_1_1_1_2"><i><font color="Green" title="E42">A16</font></i></a>, <a class="ref" href="aff_1.html#T71" title="AFF_1:th.71">AFF_1:71</a></i>;<br/><a NAME="E5:32_1_1_1_1_2"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">b'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:32"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="aff_4.html#E5:32"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="aff_4.html#E4:32_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="aff_4.html#E5:32_1"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="aff_4.html#E3:32_1_1_1_1_2"><i><font color="Green" title="E42">A16</font></i></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></i>;<br/><b>hence </b><a NAME="E6:32_1_1_1_1_2"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:32_1_1_1_1"><i><font color="Green" title="E39">A13</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="E7:32_1_1_1_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E5:32_1_1_1_1"><i><font color="Green" title="E41">A15</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="E3:32_1_1_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:32_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0: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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_1_2"/><i><font color="Green" title="E37">A17</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_1_2_1"/><i><font color="Green" title="E38">A18</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font></span><br/><b>reconsider </b><font color="Maroon" title="c11">b'</font> = <font color="Maroon" title="c10">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> <b>by </b><i><a class="txt" href="aff_4.html#E1:32_1_1_2"><i><font color="Green" title="E37">A17</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c12">M</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">b'</font>;<br/><a NAME="E3:32_1_1_2_1"/><i><font color="Green" title="E39">A19</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">b'</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c9">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="c9">a'</font>,<font color="Maroon" title="c11">b'</font> &amp; <font color="Maroon" title="c11">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="c9">a'</font>,<font color="Maroon" title="c11">b'</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:32_1_1_2_1"><i><font color="Green" title="E38">A18</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/><a NAME="E4:32_1_1_2_1"/><i><font color="Green" title="E40">A20</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">b'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:32"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="aff_4.html#E5:32_1"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="aff_4.html#E1:32_1_1_2"><i><font color="Green" title="E37">A17</font></i></a>, <a class="txt" href="aff_4.html#E1:32_1_1_2_1"><i><font color="Green" title="E38">A18</font></i></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></i>;<br/><div><i><font color="Green" title="E41">A21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_1_2_1_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">b'</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="c2">a</font>,<font color="Maroon" title="c3">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="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font></span><br/><a NAME="E2:32_1_1_2_1_1"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">b'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:32"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="aff_4.html#E5:32"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="aff_4.html#E5:32_1"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="aff_4.html#E3:32_1_1_2_1"><i><font color="Green" title="E39">A19</font></i></a>, <a class="ref" href="aff_4.html#T23" target="_self" title="AFF_4:th.23">Th23</a></i>;<br/><b>hence </b><a NAME="E3:32_1_1_2_1_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:32_1_1_2_1"><i><font color="Green" title="E39">A19</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="32_1_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_1_2_1_2"/>
not  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">b'</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="c2">a</font>,<font color="Maroon" title="c3">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="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font></span><br/><b>then consider </b><font color="Maroon" title="c13">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E3:32_1_1_2_1_2"/><i><font color="Green" title="E42">A22</font></i>: 
( <font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">b'</font> &amp; <font color="Maroon" title="c13">q</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="c2">a</font>,<font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:32"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="aff_4.html#E4:32"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="aff_4.html#E5:32"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="aff_4.html#E3:32_1_1_2_1"><i><font color="Green" title="E39">A19</font></i></a>, <a class="txt" href="aff_4.html#E4:32_1_1_2_1"><i><font color="Green" title="E40">A20</font></i></a>, <a class="ref" href="aff_4.html#T22" target="_self" title="AFF_4:th.22">Th22</a></i>;<br/><a NAME="E4:32_1_1_2_1_2"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c13">q</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E4:32_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="aff_4.html#E3:32_1_1_2_1"><i><font color="Green" title="E39">A19</font></i></a>, <a class="txt" href="aff_4.html#E3:32_1_1_2_1_2"><i><font color="Green" title="E42">A22</font></i></a>, <a class="ref" href="aff_1.html#T71" title="AFF_1:th.71">AFF_1:71</a></i>;<br/><a NAME="E5:32_1_1_2_1_2"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">b'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:32"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="aff_4.html#E5:32"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="aff_4.html#E4:32_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="aff_4.html#E5:32_1"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="aff_4.html#E3:32_1_1_2_1_2"><i><font color="Green" title="E42">A22</font></i></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></i>;<br/><b>hence </b><a NAME="E6:32_1_1_2_1_2"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:32_1_1_2_1"><i><font color="Green" title="E39">A19</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="E7:32_1_1_2_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E5:32_1_1_2_1"><i><font color="Green" title="E41">A21</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="E3:32_1_1_2"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:32_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0: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>
<b>hence </b><a NAME="E3:32_1_1"/>
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> iff <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:32_1_1"><i><font color="Green" title="E36">A10</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="E8:32_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E2:32"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</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="E8:32"/>
<font color="Maroon" title="c4">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 <b>by </b><i><a class="txt" href="aff_4.html#E4:32"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="aff_4.html#E6:32"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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>
