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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font>, <font color="Maroon" title="c4">q1</font>, <font color="Maroon" title="c5">q2</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><a NAME="E1:16"/><i><font color="Green" title="E12">A1</font></i>: 
( <font color="Maroon" title="c4">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 ;<br/>

<a NAME="E2:16"/><b>then </b><i><font color="Green" title="E13">A2</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c3">p2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T11" target="_self" title="TOPREAL1:th.11">Th11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1"/>
( <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font> or <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c3">p2</font> )
 </span><b>by </b><i><a class="txt" href="topreal1.html#E1:16"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="topreal1.html#E2:16"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_1"/>
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font>
 ;<br/><div class="add"><a NAME="E2:16_1_1_1"/><b>then </b><i><font color="Green" title="E14">A3</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font>
 <b>by </b><i><a class="txt" href="topreal1.html#E14"><i><font color="Green" title="E9">Lm2</font></i></a></i>;<br/><a NAME="E3:16_1_1_1"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <b>by </b><i><a class="txt" href="topreal1.html#E1:16"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="topreal1.html#E14"><i><font color="Green" title="E9">Lm2</font></i></a></i>;<br/><b>hence </b><a NAME="E4:16_1_1_1"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <b>by </b><i><a class="txt" href="topreal1.html#E2:16_1_1_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_2"/>
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c3">p2</font>
 ;<br/><div class="add"><a NAME="E2:16_1_1_2"/><b>then </b><i><font color="Green" title="E14">A4</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c3">p2</font>
 <b>by </b><i><a class="txt" href="topreal1.html#E14"><i><font color="Green" title="E9">Lm2</font></i></a></i>;<br/><a NAME="E3:16_1_1_2"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c3">p2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <b>by </b><i><a class="txt" href="topreal1.html#E1:16"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="topreal1.html#E14"><i><font color="Green" title="E9">Lm2</font></i></a></i>;<br/><b>hence </b><a NAME="E4:16_1_1_2"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <b>by </b><i><a class="txt" href="topreal1.html#E2:16_1_1_2"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:16"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ;<br/>


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