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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:123"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="euclidlp.html#R7" target="_self">are_coplane</a> 
 ;<br/>

<b>assume </b><a NAME="E2:123"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="euclidlp.html#R6" target="_self">_|_</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:123"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T80" target="_self">Th80</a></i>;<br/>
<a NAME="E4:123"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E2:123"><i><font color="Green">E102</font></i></a>, <a class="ref" href="euclidlp.html#T72" target="_self">Th72</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:123"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E3:123"><i><font color="Green">E103</font></i></a>, <a class="txt" href="euclidlp.html#E4:123"><i><font color="Green">E104</font></i></a>, <a class="ref" href="euclidlp.html#T84" target="_self">Th84</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:123"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="euclidlp.html#R6" target="_self">_|_</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E4:123"><i><font color="Green">E104</font></i></a>, <a class="txt" href="euclidlp.html#E6:123"><i><font color="Green">E105</font></i></a>, <a class="ref" href="euclidlp.html#T67" target="_self">Th67</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:123"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E8:123"><i><font color="Green">E106</font></i></a>, <a class="ref" href="euclidlp.html#T54" target="_self">Th54</a></i>;<br/>
<a NAME="E11:123"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></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">E108</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:123_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:123"><i><font color="Green">E101</font></i></a>, <a class="ref" href="euclidlp.html#T101" target="_self">Th101</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:123_1"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> is <a href="euclidlp.html#V1" target="_self">being_plane</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E8:123"><i><font color="Green">E106</font></i></a>, <a class="ref" href="euclidlp.html#T112" target="_self">Th112</a></i>;<br/>
<a NAME="E6:123_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E6:123"><i><font color="Green">E105</font></i></a>, <a class="txt" href="euclidlp.html#E4:123"><i><font color="Green">E104</font></i></a>, <a class="txt" href="euclidlp.html#E3:123_1"><i><font color="Green">E109</font></i></a>, <a class="txt" href="euclidlp.html#E5:123_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="euclidlp.html#E8:123"><i><font color="Green">E106</font></i></a>, <a class="ref" href="euclidlp.html#T107" target="_self">Th107</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:123_1"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="euclidlp.html#R6" target="_self">_|_</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="euclidlp.html#R5" target="_self">//</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E2:123"><i><font color="Green">E102</font></i></a>, <a class="txt" href="euclidlp.html#E10:123"><i><font color="Green">E107</font></i></a>, <a class="ref" href="euclidlp.html#T85" target="_self">Th85</a></i>;<br/>
<a NAME="E9:123_1"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E8:123_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="euclidlp.html#T71" target="_self">Th71</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></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">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:123_1"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> is <a href="euclidlp.html#V1" target="_self">being_plane</a> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:123_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="euclidlp.html#E8:123_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="euclidlp.html#T111" target="_self">Th111</a></i>;<br/>
<a NAME="E12:123_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:123_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="euclidlp.html#E3:123_1"><i><font color="Green">E109</font></i></a>, <a class="txt" href="euclidlp.html#E10:123"><i><font color="Green">E107</font></i></a>, <a class="txt" href="euclidlp.html#E6:123_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="euclidlp.html#E9:123_1"><i><font color="Green">E113</font></i></a>, <a class="txt" href="euclidlp.html#E11:123_1"><i><font color="Green">E114</font></i></a>, <a class="txt" href="euclidlp.html#E8:123_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="euclidlp.html#T107" target="_self">Th107</a></i>;<br/>
<a NAME="E13:123_1"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E8:123"><i><font color="Green">E106</font></i></a>, <a class="txt" href="euclidlp.html#E10:123"><i><font color="Green">E107</font></i></a>, <a class="txt" href="euclidlp.html#E8:123_1"><i><font color="Green">E112</font></i></a>, <a class="txt" href="euclidlp.html#E5:123_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="euclidlp.html#E11:123_1"><i><font color="Green">E114</font></i></a>, <a class="txt" href="euclidlp.html#E12:123_1"><i><font color="Green">E115</font></i></a>, <a class="ref" href="euclidlp.html#T113" target="_self">Th113</a></i>;<br/>
<a NAME="E14:123_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E6:123"><i><font color="Green">E105</font></i></a>, <a class="txt" href="euclidlp.html#E8:123"><i><font color="Green">E106</font></i></a>, <a class="ref" href="euclidlp.html#T54" target="_self">Th54</a></i>;<br/>
<b>hence </b><a NAME="E15:123_1"/>
contradiction
 <b>by </b><i><a class="txt" href="euclidlp.html#E13:123_1"><i><font color="Green">E116</font></i></a>, <a class="txt" href="euclidlp.html#E8:123_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="euclidlp.html#T76" target="_self">Th76</a>, <a class="txt" href="euclidlp.html#E1:123_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">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E10:123"><i><font color="Green">E107</font></i></a>, <a class="ref" href="euclidlp.html#T54" target="_self">Th54</a></i>;<br/>


</div>
