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<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="pre_topc.html#M1">SubSpace</a> of  <a href="topmetr.html#K3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:26"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="topmetr.html#K3">R^1</a> 
 ;<br/>

<a NAME="E2:26"/><i><font color="Green">E20</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="txt" href="topalg_2.html#E1:26"><i><font color="Green">E19</font></i></a>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="topalg_2.html#E1:26"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E4:26"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="topalg_2.html#V2" target="_self">convex</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K3">R^1</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="topalg_2.html#D3" target="_self">TOPALG_2:def 3</a><br/>

<b>thus </b><a NAME="E1:26_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> implies <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="topalg_2.html#E2:26"><i><font color="Green">E20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:26"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> is  <a href="topalg_2.html#V2" target="_self">convex</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="topalg_2.html#D4" target="_self">TOPALG_2:def 4</a><br/>


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