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<b>let </b><font color="Maroon">c<sub>1</sub></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> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:94"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="jordan2c.html#V1">Bounded</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E3:94"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="topreal6.html#T83" target="_self">Th83</a></i>;<br/>
<a NAME="E4:94"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>1</sub></font> is <a href="jordan2c.html#V1">Bounded</a>
 
<b>by </b><i><a class="txt" href="topreal6.html#E1:94"><i><font color="Green">E49</font></i></a>, <a class="ref" href="topreal6.html#T71" target="_self">Th71</a></i>;<br/>
<a NAME="E5:94"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>1</sub></font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="ref" href="topreal6.html#T88" target="_self">Th88</a></i>;<br/>
<a NAME="E6:94"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="txt" href="topreal6.html#E3:94"><i><font color="Green">E50</font></i></a>, <a class="ref" href="weierstr.html#T15">WEIERSTR:15</a></i>;<br/>
<a NAME="E7:94"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="compts_1.html#T16">COMPTS_1:16</a></i>;<br/>
<a NAME="E8:94"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="rcomp_1.html#V2">closed</a>
 
<b>by </b><i><a class="ref" href="topreal6.html#T79" target="_self">Th79</a></i>;<br/>
<a NAME="E9:94"/>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E10:94"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<b>hence </b><a NAME="E11:94"/>
 <a href="pscomp_1.html#K6">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="topreal6.html#E8:94"><i><font color="Green">E51</font></i></a>, <a class="ref" href="pscomp_1.html#T32">PSCOMP_1:32</a></i>;<br/>


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