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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="E1:111"/>
<font color="Maroon" title="c2">L1</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Maroon" title="c3">L2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font>, <font color="Maroon" title="c6">y1</font>, <font color="Maroon" title="c7">y2</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="E3:111"/><i><font color="Green" title="E88">A1</font></i>: 
( <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">x1</font>,<font color="Maroon" title="c5">x2</font> &amp; <font color="Maroon" title="c3">L2</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="c6">y1</font>,<font color="Maroon" title="c7">y2</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">x1</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c7">y2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#D7" target="_self" title="EUCLIDLP:def.7">Def7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">a</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:111"/><i><font color="Green" title="E89">A2</font></i>: 
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c5">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E3:111"><i><font color="Green" title="E88">A1</font></i></a>, <a class="ref" href="euclidlp.html#T37" target="_self" title="EUCLIDLP:th.37">Th37</a></i>;<br/>
<i><font color="Green" title="E90">A3</font></i>: <font color="Maroon" title="c5">x2</font> = 
<font color="Maroon" title="c4">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E5:111"><i><font color="Green" title="E89">A2</font></i></a>, <a class="ref" href="euclidlp.html#T11" target="_self" title="EUCLIDLP:th.11">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T3" title="EUCLID_4:th.3">EUCLID_4:3</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">y2</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T17" target="_self" title="EUCLIDLP:th.17">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">y2</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T32" title="RVSUM_1:th.32">RVSUM_1:32</a></i>
;<br/>
<a NAME="E7:111"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
;<br/>
<a NAME="E8:111"/><b>then </b><i><font color="Green" title="E91">A4</font></i>: 
<font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">y2</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E6:111"><i><font color="Green" title="E90">A3</font></i></a></i>;<br/>
<a NAME="E9:111"/>
( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">y2</font> &amp; <font color="Maroon" title="c6">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">y2</font> &amp; <font color="Maroon" title="c7">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">y2</font> )
 
<b>by </b><i><a class="ref" href="euclidlp.html#T87" target="_self" title="EUCLIDLP:th.87">Th87</a></i>;<br/>
<a NAME="E10:111"/><b>then </b>
( <font color="Maroon" title="c2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">y2</font> &amp; <font color="Maroon" title="c3">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">y2</font> )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E3:111"><i><font color="Green" title="E88">A1</font></i></a>, <a class="txt" href="euclidlp.html#E8:111"><i><font color="Green" title="E91">A4</font></i></a>, <a class="ref" href="euclidlp.html#T90" target="_self" title="EUCLIDLP:th.90">Th90</a></i>;<br/>
<b>hence </b><a NAME="E11:111"/>
<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> 
 <b>by </b><i><a class="ref" href="euclidlp.html#D12" target="_self" title="EUCLIDLP:def.12">Def12</a></i>;<br/>


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