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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">L1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="euclidlp.html#K1" target="_self">line_of_REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">L2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="euclidlp.html#K1" target="_self">line_of_REAL</a> <font color="Maroon">n</font>;<br/>





<b>assume </b><a NAME="E1:123"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">L1</font>,<font color="Maroon">L2</font> <a href="euclidlp.html#R7" target="_self">are_coplane</a> 
 ;<br/>

<b>assume </b><a NAME="E2:123"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">L1</font> <a href="euclidlp.html#R6" target="_self">_|_</a> <font color="Maroon">L2</font>
 ;<br/>

<a NAME="E3:123"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">L1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">L2</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E4:123"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">L1</font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  &amp; <font color="Maroon">L2</font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  )
 
<b>by </b><i>, <a class="txt" href="euclidlp.html#E15"><i><font color="Green">Lemma69</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">x0</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E6:123"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> &amp; not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> )
 <b>by </b><i>, , <a class="ref" href="euclidlp.html#T22" target="_self">Th22</a></i>;<br/>
<b>consider </b><font color="Maroon">L</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="euclidlp.html#K1" target="_self">line_of_REAL</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E8:123"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> &amp; <font color="Maroon">L</font> <a href="euclidlp.html#R6" target="_self">_|_</a> <font color="Maroon">L2</font> &amp; <font color="Maroon">L</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">L2</font> )
 <b>by </b><i>, <a class="txt" href="euclidlp.html#E8:123"><i><font color="Green">E105</font></i></a>, <a class="ref" href="euclidlp.html#T10" target="_self">Th10</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E10:123"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E10:123"><i><font color="Green">E106</font></i></a>, <a class="ref" href="euclidlp.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E11:123"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">E107</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font>
 ;<br/>

<b>consider </b><font color="Maroon">P0</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="euclidlp.html#K5" target="_self">plane_of_REAL</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:123_1"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">L1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P0</font> &amp; <font color="Maroon">L2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P0</font> )
 <b>by </b><i>, <a class="txt" href="euclidlp.html#E6:123"><i><font color="Green">E90</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">P</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="euclidlp.html#K5" target="_self">plane_of_REAL</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E5:123_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">P</font> is <a href="euclidlp.html#V1" target="_self">being_plane</a> &amp; <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp; <font color="Maroon">L2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E10:123"><i><font color="Green">E106</font></i></a>, <a class="txt" href="euclidlp.html#E43"><i><font color="Green">Lemma98</font></i></a></i>;<br/>
<a NAME="E6:123_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P0</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E8:123"><i><font color="Green">E105</font></i></a>, , <a class="txt" href="euclidlp.html#E5:123_1"><i><font color="Green">E109</font></i></a>, <a class="txt" href="euclidlp.html#E6:123_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="euclidlp.html#E10:123"><i><font color="Green">E106</font></i></a>, <a class="ref" href="euclidlp.html#T38" target="_self">Th38</a></i>;<br/>
<b>consider </b><font color="Maroon">L0</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="euclidlp.html#K1" target="_self">line_of_REAL</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E8:123_1"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L0</font> &amp; <font color="Maroon">L0</font> <a href="euclidlp.html#R6" target="_self">_|_</a> <font color="Maroon">L2</font> &amp; <font color="Maroon">L0</font> <a href="euclidlp.html#R5" target="_self">//</a> <font color="Maroon">L1</font> )
 <b>by </b><i>, <a class="txt" href="euclidlp.html#E1:123_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="euclidlp.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E9:123_1"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">L0</font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  &amp; <font color="Maroon">L1</font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E9:123_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="euclidlp.html#T13" target="_self">Th13</a></i>;<br/>
<b>consider </b><font color="Maroon">P1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="euclidlp.html#K5" target="_self">plane_of_REAL</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E11:123_1"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">L0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">L1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">P1</font> is <a href="euclidlp.html#V1" target="_self">being_plane</a> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E3:123_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="euclidlp.html#E9:123_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="euclidlp.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E12:123_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E3:123_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="euclidlp.html#E5:123_1"><i><font color="Green">E109</font></i></a>, <a class="txt" href="euclidlp.html#E1:123_1"><i><font color="Green">E107</font></i></a>, <a class="txt" href="euclidlp.html#E8:123_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="euclidlp.html#E11:123_1"><i><font color="Green">E113</font></i></a>, <a class="txt" href="euclidlp.html#E12:123_1"><i><font color="Green">E114</font></i></a>, <a class="txt" href="euclidlp.html#E9:123_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="euclidlp.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E13:123_1"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L0</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E10:123"><i><font color="Green">E106</font></i></a>, <a class="txt" href="euclidlp.html#E1:123_1"><i><font color="Green">E107</font></i></a>, <a class="txt" href="euclidlp.html#E9:123_1"><i><font color="Green">E112</font></i></a>, <a class="txt" href="euclidlp.html#E6:123_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="euclidlp.html#E12:123_1"><i><font color="Green">E114</font></i></a>, <a class="ref" href="euclidlp.html#T42" target="_self">Th42</a>, <a class="ref" href="euclidlp.html#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E14:123_1"/>
<font color="Maroon">L1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E8:123"><i><font color="Green">E105</font></i></a>, <a class="txt" href="euclidlp.html#E10:123"><i><font color="Green">E106</font></i></a>, <a class="ref" href="euclidlp.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E15:123_1"/>
contradiction
 <b>by </b><i><a class="ref" href="euclidlp.html#T45" target="_self">Th45</a>, <a class="txt" href="euclidlp.html#E9:123_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="euclidlp.html#T15" target="_self">Th15</a>, <a class="txt" href="euclidlp.html#E3:123_1"><i><font color="Green">E108</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:123"/>
<font color="Maroon">L1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">L2</font>
 <b>by </b><i>, <a class="ref" href="euclidlp.html#T3" target="_self">Th3</a></i>;<br/>


</div>
