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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="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:66"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="euclid_4.html#NR3" title="EUCLID_4:NR.3">is_line</a> 
 <b>and </b><br/><a NAME="E2:66"/><i><font color="Green" title="E45">A2</font></i>: 
( <font color="Maroon" title="c3">p1</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">p2</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">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p2</font> )
 ;<br/>

<a NAME="E3:66"/><i><font color="Green" title="E46">A3</font></i>: 
 ex <font color="Olive" title="b1">q1</font>, <font color="Olive" title="b2">q2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">q2</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Olive" title="b1">q1</font>,<font color="Olive" title="b2">q2</font> )
 
<b>by </b><i><a class="txt" href="euclid_4.html#E1:66"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="euclid_4.html#D8" target="_self" title="EUCLID_4:def.8">Def8</a></i>;<br/>
<a NAME="E4:66"/><b>then </b><i><font color="Green" title="E47">A4</font></i>: 
 <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</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="euclid_4.html#E2:66"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="euclid_4.html#T48" target="_self" title="EUCLID_4:th.48">Th48</a></i>;<br/>
<a NAME="E5:66"/>
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>
 
<b>by </b><i><a class="txt" href="euclid_4.html#E2:66"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="euclid_4.html#E3:66"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="euclid_4.html#T49" target="_self" title="EUCLID_4:th.49">Th49</a></i>;<br/>
<b>hence </b><a NAME="E6:66"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>
 <b>by </b><i><a class="txt" href="euclid_4.html#E4:66"><i><font color="Green" title="E47">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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