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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a> ;<br/>





<b>assume </b><a NAME="E1:88"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>thus </b><a NAME="E2:88"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="pscomp_1.html#V9">continuous</a> implies <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V5">continuous</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:88_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  st <font color="Olive">b<sub>1</sub></font> is <a href="rcomp_1.html#V2">closed</a> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pscomp_1.html#D25">PSCOMP_1:def 25</a><br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <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="pre_topc.html#D12">PRE_TOPC:def 12</a><br/>

<b>assume </b><a NAME="E2:88_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E4:88_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="rcomp_1.html#V2">closed</a>
 
<b>by </b><i><a class="txt" href="topreal6.html#E2:88_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="topreal6.html#T79" target="_self">Th79</a></i>;<br/>
<b>hence </b><a NAME="E5:88_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <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="topreal6.html#E1:88"><i><font color="Green">E44</font></i></a>, <a class="txt" href="topreal6.html#E1:88_1"><i><font color="Green">E45</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:88"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a><br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pscomp_1.html#D25">PSCOMP_1:def 25</a><br/>

<b>assume </b><a NAME="E4:88"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="rcomp_1.html#V2">closed</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</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="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E6:88"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="topreal6.html#E4:88"><i><font color="Green">E46</font></i></a>, <a class="ref" href="topreal6.html#T79" target="_self">Th79</a></i>;<br/>
<b>hence </b><a NAME="E7:88"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <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="topreal6.html#E1:88"><i><font color="Green">E44</font></i></a>, <a class="txt" href="topreal6.html#E3:88"><i><font color="Green">E45</font></i></a></i>;<br/>


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