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

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">L1</font>, <font color="Maroon" title="c3">L2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K1" title="EUCLIDLP:func.1">line_of_REAL</a> <font color="Maroon" title="c1">n</font>;<br/>





<b>assume </b><a NAME="E1:127"/><i><font color="Green" title="E105">A1</font></i>: 
( <font color="Maroon" title="c2">L1</font>,<font color="Maroon" title="c3">L2</font> <a href="euclidlp.html#R7" title="EUCLIDLP:pred.7">are_coplane</a>  &amp; <font color="Maroon" title="c2">L1</font> <a href="euclid_4.html#NR1" title="EUCLID_4:NR.1">is_line</a>  &amp; <font color="Maroon" title="c3">L2</font> <a href="euclid_4.html#NR1" title="EUCLID_4:NR.1">is_line</a>  )
 ;<br/>

<b>assume </b><a NAME="E2:127"/><i><font color="Green" title="E106">A2</font></i>: 
<font color="Maroon" title="c2">L1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">L2</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">x</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E4:127"/><i><font color="Green" title="E107">A3</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L1</font> &amp; not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L2</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E2:127"><i><font color="Green" title="E106">A2</font></i></a>, <a class="ref" href="euclidlp.html#T68" target="_self" title="EUCLIDLP:th.68">Th68</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">P</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K5" title="EUCLIDLP:func.5">plane_of_REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E6:127"/><i><font color="Green" title="E108">A4</font></i>: 
( <font color="Maroon" title="c2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c3">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:127"><i><font color="Green" title="E105">A1</font></i></a>, <a class="ref" href="euclidlp.html#T101" target="_self" title="EUCLIDLP:th.101">Th101</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">L'</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K1" title="EUCLIDLP:func.1">line_of_REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E8:127"/><i><font color="Green" title="E109">A5</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L'</font> &amp; <font color="Maroon" title="c6">L'</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c3">L2</font> &amp; <font color="Maroon" title="c6">L'</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">L2</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E4:127"><i><font color="Green" title="E107">A3</font></i></a>, <a class="txt" href="euclidlp.html#E1:127"><i><font color="Green" title="E105">A1</font></i></a>, <a class="ref" href="euclidlp.html#T67" target="_self" title="EUCLIDLP:th.67">Th67</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">x2</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E10:127"/><i><font color="Green" title="E110">A6</font></i>: 
( <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L'</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L2</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E8:127"><i><font color="Green" title="E109">A5</font></i></a>, <a class="ref" href="euclidlp.html#T54" target="_self" title="EUCLIDLP:th.54">Th54</a></i>;<br/>
<a NAME="E11:127"/><i><font color="Green" title="E111">A7</font></i>: 
( <font color="Maroon" title="c6">L'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c6">L'</font> <a href="euclid_4.html#NR1" title="EUCLID_4:NR.1">is_line</a>  )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E4:127"><i><font color="Green" title="E107">A3</font></i></a>, <a class="txt" href="euclidlp.html#E10:127"><i><font color="Green" title="E110">A6</font></i></a>, <a class="txt" href="euclidlp.html#E8:127"><i><font color="Green" title="E109">A5</font></i></a>, <a class="ref" href="euclidlp.html#T69" target="_self" title="EUCLIDLP:th.69">Th69</a></i>;<br/>
<a NAME="E12:127"/><i><font color="Green" title="E112">A8</font></i>: 
<font color="Maroon" title="c6">L'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E4:127"><i><font color="Green" title="E107">A3</font></i></a>, <a class="txt" href="euclidlp.html#E10:127"><i><font color="Green" title="E110">A6</font></i></a>, <a class="txt" href="euclidlp.html#E6:127"><i><font color="Green" title="E108">A4</font></i></a>, <a class="txt" href="euclidlp.html#E11:127"><i><font color="Green" title="E111">A7</font></i></a>, <a class="ref" href="euclidlp.html#T100" target="_self" title="EUCLIDLP:th.100">Th100</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">L0</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K1" title="EUCLIDLP:func.1">line_of_REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E14:127"/><i><font color="Green" title="E113">A9</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">L0</font> &amp; <font color="Maroon" title="c8">L0</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c6">L'</font> &amp; <font color="Maroon" title="c8">L0</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Maroon" title="c3">L2</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E8:127"><i><font color="Green" title="E109">A5</font></i></a>, <a class="ref" href="euclidlp.html#T85" target="_self" title="EUCLIDLP:th.85">Th85</a></i>;<br/>
<a NAME="E15:127"/><i><font color="Green" title="E114">A10</font></i>: 
<font color="Maroon" title="c8">L0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E14:127"><i><font color="Green" title="E113">A9</font></i></a>, <a class="txt" href="euclidlp.html#E6:127"><i><font color="Green" title="E108">A4</font></i></a>, <a class="txt" href="euclidlp.html#E8:127"><i><font color="Green" title="E109">A5</font></i></a>, <a class="txt" href="euclidlp.html#E12:127"><i><font color="Green" title="E112">A8</font></i></a>, <a class="ref" href="euclidlp.html#T115" target="_self" title="EUCLIDLP:th.115">Th115</a></i>;<br/>
<a NAME="E16:127"/><i><font color="Green" title="E115">A11</font></i>: 
<font color="Maroon" title="c8">L0</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">L1</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E4:127"><i><font color="Green" title="E107">A3</font></i></a>, <a class="txt" href="euclidlp.html#E14:127"><i><font color="Green" title="E113">A9</font></i></a>, <a class="ref" href="euclidlp.html#T54" target="_self" title="EUCLIDLP:th.54">Th54</a></i>;<br/>
<a NAME="E17:127"/><i><font color="Green" title="E116">A12</font></i>: 
( <font color="Maroon" title="c8">L0</font> <a href="euclid_4.html#NR1" title="EUCLID_4:NR.1">is_line</a>  &amp; <font color="Maroon" title="c6">L'</font> <a href="euclid_4.html#NR1" title="EUCLID_4:NR.1">is_line</a>  )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E14:127"><i><font color="Green" title="E113">A9</font></i></a>, <a class="ref" href="euclidlp.html#T72" target="_self" title="EUCLIDLP:th.72">Th72</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">x1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E19:127"/><i><font color="Green" title="E117">A13</font></i>: 
( <font color="Maroon" title="c9">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L1</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:127"><i><font color="Green" title="E105">A1</font></i></a>, <a class="txt" href="euclidlp.html#E4:127"><i><font color="Green" title="E107">A3</font></i></a>, <a class="ref" href="euclidlp.html#T58" target="_self" title="EUCLIDLP:th.58">Th58</a></i>;<br/>
<a NAME="E20:127"/><i><font color="Green" title="E118">A14</font></i>: 
<font color="Maroon" title="c2">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c9">x1</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E19:127"><i><font color="Green" title="E117">A13</font></i></a>, <a class="txt" href="euclidlp.html#E4:127"><i><font color="Green" title="E107">A3</font></i></a>, <a class="ref" href="euclidlp.html#T69" target="_self" title="EUCLIDLP:th.69">Th69</a></i>;<br/>
<a NAME="E21:127"/>
<font color="Maroon" title="c2">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:127_1"/><i><font color="Green" title="E119">A15</font></i>: 
<font color="Maroon" title="c2">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">L0</font>
 ;<br/>

<a NAME="E2:127_1"/><i><font color="Green" title="E120">A16</font></i>: 
not <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L'</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E8:127"><i><font color="Green" title="E109">A5</font></i></a>, <a class="txt" href="euclidlp.html#E19:127"><i><font color="Green" title="E117">A13</font></i></a>, <a class="ref" href="euclidlp.html#T69" target="_self" title="EUCLIDLP:th.69">Th69</a>, <a class="txt" href="euclidlp.html#E20:127"><i><font color="Green" title="E118">A14</font></i></a>, <a class="txt" href="euclidlp.html#E2:127"><i><font color="Green" title="E106">A2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">L</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K1" title="EUCLIDLP:func.1">line_of_REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E4:127_1"/><i><font color="Green" title="E121">A17</font></i>: 
( <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">L</font> &amp; <font color="Maroon" title="c6">L'</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c10">L</font> &amp; <font color="Maroon" title="c6">L'</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c10">L</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E17:127"><i><font color="Green" title="E116">A12</font></i></a>, <a class="ref" href="euclidlp.html#T67" target="_self" title="EUCLIDLP:th.67">Th67</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">x'</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E6:127_1"/><i><font color="Green" title="E122">A18</font></i>: 
( <font color="Maroon" title="c11">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L'</font> &amp; <font color="Maroon" title="c11">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">L</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E4:127_1"><i><font color="Green" title="E121">A17</font></i></a>, <a class="ref" href="euclidlp.html#T54" target="_self" title="EUCLIDLP:th.54">Th54</a></i>;<br/>
<a NAME="E7:127_1"/><i><font color="Green" title="E123">A19</font></i>: 
<font color="Maroon" title="c10">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c11">x'</font>,<font color="Maroon" title="c9">x1</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E6:127_1"><i><font color="Green" title="E122">A18</font></i></a>, <a class="txt" href="euclidlp.html#E2:127_1"><i><font color="Green" title="E120">A16</font></i></a>, <a class="txt" href="euclidlp.html#E4:127_1"><i><font color="Green" title="E121">A17</font></i></a>, <a class="ref" href="euclidlp.html#T69" target="_self" title="EUCLIDLP:th.69">Th69</a></i>;<br/>
<a NAME="E8:127_1"/><i><font color="Green" title="E124">A20</font></i>: 
<font color="Maroon" title="c10">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E6:127_1"><i><font color="Green" title="E122">A18</font></i></a>, <a class="txt" href="euclidlp.html#E12:127"><i><font color="Green" title="E112">A8</font></i></a>, <a class="txt" href="euclidlp.html#E19:127"><i><font color="Green" title="E117">A13</font></i></a>, <a class="txt" href="euclidlp.html#E6:127"><i><font color="Green" title="E108">A4</font></i></a>, <a class="txt" href="euclidlp.html#E7:127_1"><i><font color="Green" title="E123">A19</font></i></a>, <a class="ref" href="euclidlp.html#T100" target="_self" title="EUCLIDLP:th.100">Th100</a></i>;<br/>
<a NAME="E9:127_1"/><i><font color="Green" title="E125">A21</font></i>: 
<font color="Maroon" title="c10">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">L0</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E14:127"><i><font color="Green" title="E113">A9</font></i></a>, <a class="txt" href="euclidlp.html#E19:127"><i><font color="Green" title="E117">A13</font></i></a>, <a class="ref" href="euclidlp.html#T69" target="_self" title="EUCLIDLP:th.69">Th69</a>, <a class="txt" href="euclidlp.html#E1:127_1"><i><font color="Green" title="E119">A15</font></i></a>, <a class="txt" href="euclidlp.html#E20:127"><i><font color="Green" title="E118">A14</font></i></a>, <a class="txt" href="euclidlp.html#E4:127_1"><i><font color="Green" title="E121">A17</font></i></a></i>;<br/>
<a NAME="E10:127_1"/><i><font color="Green" title="E126">A22</font></i>: 
<font color="Maroon" title="c10">L</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Maroon" title="c8">L0</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E14:127"><i><font color="Green" title="E113">A9</font></i></a>, <a class="txt" href="euclidlp.html#E12:127"><i><font color="Green" title="E112">A8</font></i></a>, <a class="txt" href="euclidlp.html#E15:127"><i><font color="Green" title="E114">A10</font></i></a>, <a class="txt" href="euclidlp.html#E4:127_1"><i><font color="Green" title="E121">A17</font></i></a>, <a class="txt" href="euclidlp.html#E8:127_1"><i><font color="Green" title="E124">A20</font></i></a>, <a class="ref" href="euclidlp.html#T116" target="_self" title="EUCLIDLP:th.116">Th116</a></i>;<br/>
<a NAME="E11:127_1"/><i><font color="Green" title="E127">A23</font></i>: 
<font color="Maroon" title="c10">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">L2</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E2:127"><i><font color="Green" title="E106">A2</font></i></a>, <a class="txt" href="euclidlp.html#E19:127"><i><font color="Green" title="E117">A13</font></i></a>, <a class="ref" href="euclidlp.html#T54" target="_self" title="EUCLIDLP:th.54">Th54</a>, <a class="txt" href="euclidlp.html#E4:127_1"><i><font color="Green" title="E121">A17</font></i></a></i>;<br/>
<a NAME="E12:127_1"/>
<font color="Maroon" title="c10">L</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">L1</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E19:127"><i><font color="Green" title="E117">A13</font></i></a>, <a class="txt" href="euclidlp.html#E4:127_1"><i><font color="Green" title="E121">A17</font></i></a>, <a class="ref" href="euclidlp.html#T54" target="_self" title="EUCLIDLP:th.54">Th54</a></i>;<br/>
<b>hence </b><a NAME="E13:127_1"/>
contradiction
 <b>by </b><i><a class="txt" href="euclidlp.html#E2:127"><i><font color="Green" title="E106">A2</font></i></a>, <a class="txt" href="euclidlp.html#E6:127"><i><font color="Green" title="E108">A4</font></i></a>, <a class="txt" href="euclidlp.html#E15:127"><i><font color="Green" title="E114">A10</font></i></a>, <a class="txt" href="euclidlp.html#E8:127_1"><i><font color="Green" title="E124">A20</font></i></a>, <a class="txt" href="euclidlp.html#E4:127"><i><font color="Green" title="E107">A3</font></i></a>, <a class="txt" href="euclidlp.html#E14:127"><i><font color="Green" title="E113">A9</font></i></a>, <a class="txt" href="euclidlp.html#E9:127_1"><i><font color="Green" title="E125">A21</font></i></a>, <a class="txt" href="euclidlp.html#E10:127_1"><i><font color="Green" title="E126">A22</font></i></a>, <a class="txt" href="euclidlp.html#E11:127_1"><i><font color="Green" title="E127">A23</font></i></a>, <a class="txt" href="euclidlp.html#E16:127"><i><font color="Green" title="E115">A11</font></i></a>, <a class="ref" href="euclidlp.html#T117" target="_self" title="EUCLIDLP:th.117">Th117</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E22:127"/>
<font color="Maroon" title="c2">L1</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Maroon" title="c3">L2</font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E14:127"><i><font color="Green" title="E113">A9</font></i></a></i>;<br/>


</div>
