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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">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">x2</font> be    <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>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:32"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="euclid_4.html#NR1" title="EUCLID_4:NR.1">is_line</a> 
 <b>and </b><br/><a NAME="E2:32"/><i><font color="Green" title="E36">A2</font></i>: 
( <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c4">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c3">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x2</font> )
 ;<br/>

<a NAME="E3:32"/><i><font color="Green" title="E37">A3</font></i>: 
 ex <font color="Olive" title="b1">y1</font>, <font color="Olive" title="b2">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> st <br/>( <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y2</font> &amp; <font color="Maroon" title="c2">A</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="Olive" title="b1">y1</font>,<font color="Olive" title="b2">y2</font> )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:32"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="euclid_4.html#D2" title="EUCLID_4:def.2">EUCLID_4:def 2</a></i>;<br/>
<a NAME="E4:32"/><b>then </b><i><font color="Green" title="E38">A4</font></i>: 
 <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E2:32"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="euclid_4.html#T11" title="EUCLID_4:th.11">EUCLID_4:11</a></i>;<br/>
<a NAME="E5:32"/>
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E2:32"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="euclidlp.html#E3:32"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="euclid_4.html#T12" title="EUCLID_4:th.12">EUCLID_4:12</a></i>;<br/>
<b>hence </b><a NAME="E6:32"/>
<font color="Maroon" title="c2">A</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="c3">x1</font>,<font color="Maroon" title="c4">x2</font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E4:32"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


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