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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:36"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:36"/>
for <font color="Olive">p1</font>, <font color="Olive">p2</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>  st <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/> <a href="topreal1.html#K3">LSeg</a> <font color="Olive">p1</font>,<font color="Olive">p2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:36_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 ;<br/>

<a NAME="E2:36_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<b>hence </b><a NAME="E3:36_1"/>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:36"/><b>then </b>
<font color="Maroon">P</font> is <a href="jordan1.html#V1">convex</a>
 
<b>by </b><i><a class="ref" href="jordan1.html#D1">JORDAN1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:36"/>
<font color="Maroon">P</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="jordan1.html#T9">JORDAN1:9</a></i>;<br/>


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