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<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="E1:123"/><i><font color="Green" title="E97">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> 
 ;<br/>

<b>assume </b><a NAME="E2:123"/><i><font color="Green" title="E98">A2</font></i>: 
<font color="Maroon" title="c2">L1</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c3">L2</font>
 ;<br/>

<a NAME="E3:123"/><b>then </b><i><font color="Green" title="E99">A3</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="c3">L2</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T80" target="_self" title="EUCLIDLP:th.80">Th80</a></i>;<br/>
<a NAME="E4:123"/><i><font color="Green" title="E100">A4</font></i>: 
( <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>  )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E2:123"><i><font color="Green" title="E98">A2</font></i></a>, <a class="ref" href="euclidlp.html#T72" target="_self" title="EUCLIDLP:th.72">Th72</a></i>;<br/>
<b>then consider </b><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><b> such that </b><br/><a NAME="E6:123"/><i><font color="Green" title="E101">A5</font></i>: 
( <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> &amp; 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> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E3:123"><i><font color="Green" title="E99">A3</font></i></a>, <a class="ref" href="euclidlp.html#T84" target="_self" title="EUCLIDLP:th.84">Th84</a></i>;<br/>
<b>consider </b><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><b> such that </b><br/><a NAME="E8:123"/><i><font color="Green" title="E102">A6</font></i>: 
( <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> &amp; <font color="Maroon" title="c5">L</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c3">L2</font> &amp; <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> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E4:123"><i><font color="Green" title="E100">A4</font></i></a>, <a class="txt" href="euclidlp.html#E6:123"><i><font color="Green" title="E101">A5</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="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><b> such that </b><br/><a NAME="E10:123"/><i><font color="Green" title="E103">A7</font></i>: 
( <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> &amp; <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> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E8:123"><i><font color="Green" title="E102">A6</font></i></a>, <a class="ref" href="euclidlp.html#T54" target="_self" title="EUCLIDLP:th.54">Th54</a></i>;<br/>
<a NAME="E11:123"/>
<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="123_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:123_1"/><i><font color="Green" title="E104">A8</font></i>: 
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>
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">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><b> such that </b><br/><a NAME="E3:123_1"/><i><font color="Green" title="E105">A9</font></i>: 
( <font color="Maroon" title="c2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">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="c7">P0</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:123"><i><font color="Green" title="E97">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="c8">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="E5:123_1"/><i><font color="Green" title="E106">A10</font></i>: 
( <font color="Maroon" title="c8">P</font> is <a href="euclidlp.html#V1" title="EUCLIDLP:attr.1">being_plane</a> &amp; <font color="Maroon" title="c5">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">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="c8">P</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E8:123"><i><font color="Green" title="E102">A6</font></i></a>, <a class="ref" href="euclidlp.html#T112" target="_self" title="EUCLIDLP:th.112">Th112</a></i>;<br/>
<a NAME="E6:123_1"/><i><font color="Green" title="E107">A11</font></i>: 
<font color="Maroon" title="c8">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">P0</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E4:123"><i><font color="Green" title="E100">A4</font></i></a>, <a class="txt" href="euclidlp.html#E6:123"><i><font color="Green" title="E101">A5</font></i></a>, <a class="txt" href="euclidlp.html#E8:123"><i><font color="Green" title="E102">A6</font></i></a>, <a class="txt" href="euclidlp.html#E3:123_1"><i><font color="Green" title="E105">A9</font></i></a>, <a class="txt" href="euclidlp.html#E5:123_1"><i><font color="Green" title="E106">A10</font></i></a>, <a class="ref" href="euclidlp.html#T107" target="_self" title="EUCLIDLP:th.107">Th107</a></i>;<br/>
<b>consider </b><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><b> such that </b><br/><a NAME="E8:123_1"/><i><font color="Green" title="E108">A12</font></i>: 
( <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> &amp; <font color="Maroon" title="c9">L0</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c3">L2</font> &amp; <font color="Maroon" title="c9">L0</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Maroon" title="c2">L1</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E2:123"><i><font color="Green" title="E98">A2</font></i></a>, <a class="txt" href="euclidlp.html#E10:123"><i><font color="Green" title="E103">A7</font></i></a>, <a class="ref" href="euclidlp.html#T85" target="_self" title="EUCLIDLP:th.85">Th85</a></i>;<br/>
<a NAME="E9:123_1"/><i><font color="Green" title="E109">A13</font></i>: 
( <font color="Maroon" title="c9">L0</font> <a href="euclid_4.html#NR1" title="EUCLID_4:NR.1">is_line</a>  &amp; <font color="Maroon" title="c2">L1</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#E8:123_1"><i><font color="Green" title="E108">A12</font></i></a>, <a class="ref" href="euclidlp.html#T71" target="_self" title="EUCLIDLP:th.71">Th71</a></i>;<br/>
<b>consider </b><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><b> such that </b><br/><a NAME="E11:123_1"/><i><font color="Green" title="E110">A14</font></i>: 
( <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> &amp; <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> &amp; <font color="Maroon" title="c10">P1</font> is <a href="euclidlp.html#V1" title="EUCLIDLP:attr.1">being_plane</a> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:123_1"><i><font color="Green" title="E104">A8</font></i></a>, <a class="txt" href="euclidlp.html#E8:123_1"><i><font color="Green" title="E108">A12</font></i></a>, <a class="ref" href="euclidlp.html#T111" target="_self" title="EUCLIDLP:th.111">Th111</a></i>;<br/>
<a NAME="E12:123_1"/>
<font color="Maroon" title="c8">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">P1</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E10:123"><i><font color="Green" title="E103">A7</font></i></a>, <a class="txt" href="euclidlp.html#E1:123_1"><i><font color="Green" title="E104">A8</font></i></a>, <a class="txt" href="euclidlp.html#E3:123_1"><i><font color="Green" title="E105">A9</font></i></a>, <a class="txt" href="euclidlp.html#E6:123_1"><i><font color="Green" title="E107">A11</font></i></a>, <a class="txt" href="euclidlp.html#E8:123_1"><i><font color="Green" title="E108">A12</font></i></a>, <a class="txt" href="euclidlp.html#E9:123_1"><i><font color="Green" title="E109">A13</font></i></a>, <a class="txt" href="euclidlp.html#E11:123_1"><i><font color="Green" title="E110">A14</font></i></a>, <a class="ref" href="euclidlp.html#T107" target="_self" title="EUCLIDLP:th.107">Th107</a></i>;<br/>
<a NAME="E13:123_1"/><b>then </b><i><font color="Green" title="E111">A15</font></i>: 
<font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">L0</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E8:123"><i><font color="Green" title="E102">A6</font></i></a>, <a class="txt" href="euclidlp.html#E10:123"><i><font color="Green" title="E103">A7</font></i></a>, <a class="txt" href="euclidlp.html#E5:123_1"><i><font color="Green" title="E106">A10</font></i></a>, <a class="txt" href="euclidlp.html#E8:123_1"><i><font color="Green" title="E108">A12</font></i></a>, <a class="txt" href="euclidlp.html#E11:123_1"><i><font color="Green" title="E110">A14</font></i></a>, <a class="ref" href="euclidlp.html#T113" target="_self" title="EUCLIDLP:th.113">Th113</a></i>;<br/>
<a NAME="E14:123_1"/>
<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>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E6:123"><i><font color="Green" title="E101">A5</font></i></a>, <a class="txt" href="euclidlp.html#E8:123"><i><font color="Green" title="E102">A6</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="E15:123_1"/>
contradiction
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:123_1"><i><font color="Green" title="E104">A8</font></i></a>, <a class="txt" href="euclidlp.html#E8:123_1"><i><font color="Green" title="E108">A12</font></i></a>, <a class="txt" href="euclidlp.html#E13:123_1"><i><font color="Green" title="E111">A15</font></i></a>, <a class="ref" href="euclidlp.html#T76" target="_self" title="EUCLIDLP:th.76">Th76</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:123"/>
<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>
 <b>by </b><i><a class="txt" href="euclidlp.html#E10:123"><i><font color="Green" title="E103">A7</font></i></a>, <a class="ref" href="euclidlp.html#T54" target="_self" title="EUCLIDLP:th.54">Th54</a></i>;<br/>


</div>
