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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>



<b>assume </b><a NAME="E1:55"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:55"/>
the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T35">BORSUK_1:35</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</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="borsuk_4.html#E1:55"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T35">BORSUK_1:35</a></i>;<br/>
<a NAME="E5:55"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E53"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="borsuk_1.html#D14">BORSUK_1:def 14</a></i>;<br/>
<a NAME="E6:55"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a> iff <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E1:55"><i><font color="Green">E47</font></i></a>, <a class="ref" href="tsep_1.html#T8">TSEP_1:8</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:55_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="rcomp_1.html#V2">closed</a>
 ;<br/><a NAME="E2:55_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="topreal6.html#T79">TOPREAL6:79</a></i>;<br/><a NAME="E3:55_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E1:55"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><b>hence </b><a NAME="E4:55_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E2:55_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E8:55"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<b>hence </b><a NAME="E9:55"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="rcomp_1.html#V2">closed</a>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E6:55"><i><font color="Green">E48</font></i></a>, <a class="ref" href="topreal6.html#T79">TOPREAL6:79</a></i>;<br/>


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