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<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <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/>  for <font color="Olive" title="b2">t</font>, <font color="Olive" title="b3">s1</font>, <font color="Olive" title="b4">s2</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> holds  <font color="Olive" title="b2">t</font>,<font color="Olive" title="b2">t</font>,<font color="Olive" title="b1">V</font> <a href="jordan18.html#R1" title="JORDAN18:pred.1">-separate</a> <font color="Olive" title="b3">s1</font>,<font color="Olive" title="b4">s2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">t</font>, <font color="Olive" title="b2">s1</font>, <font color="Olive" title="b3">s2</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> holds  <font color="Olive" title="b1">t</font>,<font color="Olive" title="b1">t</font>,<font color="Maroon" title="c2">V</font> <a href="jordan18.html#R1" title="JORDAN18:pred.1">-separate</a> <font color="Olive" title="b2">s1</font>,<font color="Olive" title="b3">s2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">t</font>, <font color="Maroon" title="c4">s1</font>, <font color="Maroon" title="c5">s2</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c2">V</font> <a href="jordan18.html#R1" title="JORDAN18:pred.1">-separate</a> <font color="Maroon" title="c4">s1</font>,<font color="Maroon" title="c5">s2</font></span><br/>









<span class="kw">assume </span><a NAME="E1:27"/>
 not <font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c2">V</font> <a href="jordan18.html#R1" title="JORDAN18:pred.1">-separate</a> <font color="Maroon" title="c4">s1</font>,<font color="Maroon" title="c5">s2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:27"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">A</font> being   <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> st <br/>( <font color="Olive" title="b1">A</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c3">t</font> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Olive" title="b1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">s1</font>,<font color="Maroon" title="c5">s2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan18.html#D3" target="_self" title="JORDAN18:def.3">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:27"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T37" title="JORDAN6:th.37">JORDAN6:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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