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<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>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <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>;<br/>











<b>assume </b><a NAME="E1:128"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclidlp.html#R3" target="_self">are_lindependent2</a>  )
 ;<br/>

<a NAME="E2:128"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E43"><i><font color="Green">Lemma40</font></i></a></i>;<br/>
<a NAME="E3:128"/><b>then </b><i><font color="Green">E106</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="euclidlp.html#T14" target="_self">Th14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>8</sub></font> =  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> as    <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>by </b><i><a class="ref" href="euclidlp.html#T52" target="_self">Th52</a></i>;<br/>
<a NAME="E5:128"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:128"><i><font color="Green">E105</font></i></a>, <a class="ref" href="euclidlp.html#T100" target="_self">Th100</a></i>;<br/>
<a NAME="E6:128"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="euclidlp.html#R7" target="_self">are_coplane</a> 
 
<b>by </b><i><a class="ref" href="euclidlp.html#T101" target="_self">Th101</a></i>;<br/>
<a NAME="E7:128"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E3:128"><i><font color="Green">E106</font></i></a>, <a class="ref" href="euclid_4.html#D2">EUCLID_4:def 2</a></i>;<br/>
<a NAME="E8:128"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="ref" href="euclid_4.html#T10">EUCLID_4:10</a></i>;<br/>
<a NAME="E9:128"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="128_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:128_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:128_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="euclidlp.html#R5" target="_self">//</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E6:128"><i><font color="Green">E107</font></i></a>, <a class="txt" href="euclidlp.html#E7:128"><i><font color="Green">E108</font></i></a>, <a class="ref" href="euclidlp.html#T118" target="_self">Th118</a></i>;<br/>
<a NAME="E3:128_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E3:128"><i><font color="Green">E106</font></i></a>, <a class="txt" href="euclidlp.html#E8:128"><i><font color="Green">E109</font></i></a>, <a class="ref" href="euclidlp.html#T82" target="_self">Th82</a></i>;<br/>
<b>hence </b><a NAME="E4:128_1"/>
contradiction
 <b>by </b><i><a class="txt" href="euclidlp.html#E51"><i><font color="Green">Lemma45</font></i></a>, <a class="txt" href="euclidlp.html#E1:128"><i><font color="Green">E105</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:128"/>
 <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/>


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