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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="E1:118"/><i><font color="Green" title="E97">A1</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>  &amp; <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> &amp; <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> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">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="E3:118"/><i><font color="Green" title="E98">A2</font></i>: 
( <font color="Maroon" title="c4">x</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">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="ref" href="euclidlp.html#T84" target="_self" title="EUCLIDLP:th.84">Th84</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">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="E5:118"/><i><font color="Green" title="E99">A3</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">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="c5">P</font> &amp; <font color="Maroon" title="c5">P</font> is <a href="euclidlp.html#V1" title="EUCLIDLP:attr.1">being_plane</a> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:118"><i><font color="Green" title="E97">A1</font></i></a>, <a class="txt" href="euclidlp.html#E3:118"><i><font color="Green" title="E98">A2</font></i></a>, <a class="ref" href="euclidlp.html#T105" target="_self" title="EUCLIDLP:th.105">Th105</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">y</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="E7:118"/><i><font color="Green" title="E100">A4</font></i>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L1</font> &amp; <font color="Maroon" title="c6">y</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#E1:118"><i><font color="Green" title="E97">A1</font></i></a>, <a class="ref" href="euclidlp.html#T54" target="_self" title="EUCLIDLP:th.54">Th54</a></i>;<br/>
<a NAME="E8:118"/>
<font color="Maroon" title="c2">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">y</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E7:118"><i><font color="Green" title="E100">A4</font></i></a>, <a class="txt" href="euclidlp.html#E3:118"><i><font color="Green" title="E98">A2</font></i></a>, <a class="txt" href="euclidlp.html#E1:118"><i><font color="Green" title="E97">A1</font></i></a>, <a class="ref" href="euclidlp.html#T35" target="_self" title="EUCLIDLP:th.35">Th35</a></i>;<br/>
<a NAME="E9:118"/><b>then </b>
<font color="Maroon" title="c2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E5:118"><i><font color="Green" title="E99">A3</font></i></a>, <a class="txt" href="euclidlp.html#E7:118"><i><font color="Green" title="E100">A4</font></i></a>, <a class="ref" href="euclidlp.html#T100" target="_self" title="EUCLIDLP:th.100">Th100</a></i>;<br/>
<b>hence </b><a NAME="E10:118"/>
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Maroon" title="c2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c3">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> is <a href="euclidlp.html#V1" title="EUCLIDLP:attr.1">being_plane</a> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E5:118"><i><font color="Green" title="E99">A3</font></i></a></i>;<br/>


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