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

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">L1</font>, <font color="Olive" title="b2">L2</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>  st <font color="Olive" title="b1">L1</font>,<font color="Olive" title="b2">L2</font> <a href="euclidlp.html#R7" title="EUCLIDLP:pred.7">are_coplane</a>  &amp; <font color="Olive" title="b1">L1</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Olive" title="b2">L2</font> holds <br/><font color="Olive" title="b1">L1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">L2</font></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c3">L2</font> implies <font color="Maroon" title="c2">L1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">L2</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:122"/><span class="lab"><font color="Green" title="E94">A1</font></span>: 
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">L1</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c3">L2</font> or <font color="Maroon" title="c2">L1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">L2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:122"/><span class="lab"><font color="Green" title="E95">A2</font></span>: 
<font color="Maroon" title="c2">L1</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c3">L2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">L1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">L2</font></span><br/>

<a NAME="E3:122"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A3</font></span>: 
<font color="Maroon" title="c3">L2</font> is  <a href="euclid_4.html#V1" title="EUCLID_4:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T67" target="_self" title="EUCLIDLP:th.67">Th72</a></span>;<br/>
<a NAME="E4:122"/>
<font color="Maroon" title="c2">L1</font> is  <a href="euclid_4.html#V1" title="EUCLID_4:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:122"><span class="lab"><font color="Green" title="E95">A2</font></span></a>, <a class="ref" href="euclidlp.html#T67" target="_self" title="EUCLIDLP:th.67">Th72</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">x0</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><span class="kw"> such that </span><br/><a NAME="E6:122"/><span class="lab"><font color="Green" title="E97">A4</font></span>: 
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L1</font>
 <span class="kw">and </span><br/><a NAME="E7:122"/><span class="lab"><font color="Green" title="E98">A5</font></span>: 
 not <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:122"><span class="lab"><font color="Green" title="E95">A2</font></span></a>, <a class="txt" href="#E3:122"><span class="lab"><font color="Green" title="E96">A3</font></span></a>, <a class="ref" href="euclidlp.html#T75" target="_self" title="EUCLIDLP:th.75">Th80</a>, <a class="ref" href="euclidlp.html#T79" target="_self" title="EUCLIDLP:th.79">Th84</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">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><span class="kw"> such that </span><br/><a NAME="E9:122"/><span class="lab"><font color="Green" title="E99">A6</font></span>: 
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">L</font>
 <span class="kw">and </span><br/><a NAME="E10:122"/><span class="lab"><font color="Green" title="E100">A7</font></span>: 
<font color="Maroon" title="c5">L</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c3">L2</font>
 <span class="kw">and </span><br/><a NAME="E11:122"/><span class="lab"><font color="Green" title="E101">A8</font></span>: 
<font color="Maroon" title="c5">L</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">L2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:122"><span class="lab"><font color="Green" title="E96">A3</font></span></a>, <a class="txt" href="#E7:122"><span class="lab"><font color="Green" title="E98">A5</font></span></a>, <a class="ref" href="euclidlp.html#T62" target="_self" title="EUCLIDLP:th.62">Th67</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">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><span class="kw"> such that </span><br/><a NAME="E13:122"/><span class="lab"><font color="Green" title="E102">A9</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">L</font>
 <span class="kw">and </span><br/><a NAME="E14:122"/><span class="lab"><font color="Green" title="E103">A10</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:122"><span class="lab"><font color="Green" title="E101">A8</font></span></a>, <a class="ref" href="euclidlp.html#T49" target="_self" title="EUCLIDLP:th.49">Th54</a></span>;<br/>
<a NAME="E15:122"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:122_1"/><span class="lab"><font color="Green" title="E104">A11</font></span>: 
<font color="Maroon" title="c2">L1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c5">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:122"><span class="lab"><font color="Green" title="E97">A4</font></span></a>, <a class="txt" href="#E9:122"><span class="lab"><font color="Green" title="E99">A6</font></span></a>, <a class="ref" href="euclidlp.html#T49" target="_self" title="EUCLIDLP:th.49">Th54</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">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><span class="kw"> such that </span><br/><a NAME="E3:122_1"/>
<font color="Maroon" title="c7">P</font> is  <a href="euclidlp.html#V1" title="EUCLIDLP:attr.1">being_plane</a> 
 <span class="kw">and </span><br/><a NAME="E4:122_1"/><span class="lab"><font color="Green" title="E105">A12</font></span>: 
( <font color="Maroon" title="c5">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">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="c7">P</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:122"><span class="lab"><font color="Green" title="E100">A7</font></span></a>, <a class="txt" href="#E11:122"><span class="lab"><font color="Green" title="E101">A8</font></span></a>, <a class="ref" href="euclidlp.html#T107" target="_self" title="EUCLIDLP:th.107">Th112</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">P0</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><span class="kw"> such that </span><br/><a NAME="E6:122_1"/><span class="lab"><font color="Green" title="E106">A13</font></span>: 
( <font color="Maroon" title="c2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">P0</font> &amp; <font color="Maroon" title="c3">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">P0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:122"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="ref" href="euclidlp.html#T96" target="_self" title="EUCLIDLP:th.96">Th101</a></span>;<br/>
<a NAME="E7:122_1"/><span class="lab"><font color="Green" title="E107">A14</font></span>: 
<font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:122"><span class="lab"><font color="Green" title="E96">A3</font></span></a>, <a class="txt" href="#E6:122"><span class="lab"><font color="Green" title="E97">A4</font></span></a>, <a class="txt" href="#E7:122"><span class="lab"><font color="Green" title="E98">A5</font></span></a>, <a class="txt" href="#E9:122"><span class="lab"><font color="Green" title="E99">A6</font></span></a>, <a class="txt" href="#E6:122_1"><span class="lab"><font color="Green" title="E106">A13</font></span></a>, <a class="txt" href="#E4:122_1"><span class="lab"><font color="Green" title="E105">A12</font></span></a>, <a class="ref" href="euclidlp.html#T102" target="_self" title="EUCLIDLP:th.102">Th107</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">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><span class="kw"> such that </span><br/><a NAME="E9:122_1"/><span class="lab"><font color="Green" title="E108">A15</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">L0</font>
 <span class="kw">and </span><br/><a NAME="E10:122_1"/><span class="lab"><font color="Green" title="E109">A16</font></span>: 
<font color="Maroon" title="c9">L0</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c3">L2</font>
 <span class="kw">and </span><br/><a NAME="E11:122_1"/><span class="lab"><font color="Green" title="E110">A17</font></span>: 
<font color="Maroon" title="c9">L0</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Maroon" title="c2">L1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:122"><span class="lab"><font color="Green" title="E95">A2</font></span></a>, <a class="ref" href="euclidlp.html#T80" target="_self" title="EUCLIDLP:th.80">Th85</a></span>;<br/>
<span class="kw">assume </span><a NAME="E12:122_1"/><span class="lab"><font color="Green" title="E111">A18</font></span>: 
 not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">P1</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><span class="kw"> such that </span><br/><a NAME="E14:122_1"/><span class="lab"><font color="Green" title="E112">A19</font></span>: 
<font color="Maroon" title="c9">L0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">P1</font>
 <span class="kw">and </span><br/><a NAME="E15:122_1"/><span class="lab"><font color="Green" title="E113">A20</font></span>: 
<font color="Maroon" title="c2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">P1</font>
 <span class="kw">and </span><br/><a NAME="E16:122_1"/>
<font color="Maroon" title="c10">P1</font> is  <a href="euclidlp.html#V1" title="EUCLIDLP:attr.1">being_plane</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:122_1"><span class="lab"><font color="Green" title="E108">A15</font></span></a>, <a class="txt" href="#E11:122_1"><span class="lab"><font color="Green" title="E110">A17</font></span></a>, <a class="ref" href="euclidlp.html#T106" target="_self" title="EUCLIDLP:th.106">Th111</a></span>;<br/>
<a NAME="E17:122_1"/>
<font color="Maroon" title="c2">L1</font> is  <a href="euclid_4.html#V1" title="EUCLID_4:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:122_1"><span class="lab"><font color="Green" title="E110">A17</font></span></a>, <a class="ref" href="euclidlp.html#T66" target="_self" title="EUCLIDLP:th.66">Th71</a></span>;<br/>
<a NAME="E18:122_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">P1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:122"><span class="lab"><font color="Green" title="E103">A10</font></span></a>, <a class="txt" href="#E12:122_1"><span class="lab"><font color="Green" title="E111">A18</font></span></a>, <a class="txt" href="#E6:122_1"><span class="lab"><font color="Green" title="E106">A13</font></span></a>, <a class="txt" href="#E7:122_1"><span class="lab"><font color="Green" title="E107">A14</font></span></a>, <a class="txt" href="#E9:122_1"><span class="lab"><font color="Green" title="E108">A15</font></span></a>, <a class="txt" href="#E14:122_1"><span class="lab"><font color="Green" title="E112">A19</font></span></a>, <a class="txt" href="#E15:122_1"><span class="lab"><font color="Green" title="E113">A20</font></span></a>, <a class="ref" href="euclidlp.html#T102" target="_self" title="EUCLIDLP:th.102">Th107</a></span>;<br/>
<a NAME="E19:122_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">L0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:122"><span class="lab"><font color="Green" title="E100">A7</font></span></a>, <a class="txt" href="#E13:122"><span class="lab"><font color="Green" title="E102">A9</font></span></a>, <a class="txt" href="#E14:122"><span class="lab"><font color="Green" title="E103">A10</font></span></a>, <a class="txt" href="#E4:122_1"><span class="lab"><font color="Green" title="E105">A12</font></span></a>, <a class="txt" href="#E9:122_1"><span class="lab"><font color="Green" title="E108">A15</font></span></a>, <a class="txt" href="#E10:122_1"><span class="lab"><font color="Green" title="E109">A16</font></span></a>, <a class="txt" href="#E14:122_1"><span class="lab"><font color="Green" title="E112">A19</font></span></a>, <a class="ref" href="euclidlp.html#T108" target="_self" title="EUCLIDLP:th.108">Th113</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:122_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:122_1"><span class="lab"><font color="Green" title="E111">A18</font></span></a>, <a class="txt" href="#E9:122_1"><span class="lab"><font color="Green" title="E108">A15</font></span></a>, <a class="txt" href="#E11:122_1"><span class="lab"><font color="Green" title="E110">A17</font></span></a>, <a class="txt" href="#E1:122_1"><span class="lab"><font color="Green" title="E104">A11</font></span></a>, <a class="ref" href="euclidlp.html#T71" target="_self" title="EUCLIDLP:th.71">Th76</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E16:122"/>
<font color="Maroon" title="c2">L1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">L2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:122"><span class="lab"><font color="Green" title="E103">A10</font></span></a>, <a class="ref" href="euclidlp.html#T49" target="_self" title="EUCLIDLP:th.49">Th54</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
