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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/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:88_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="pscomp_1.html#V9" target="_self">continuous</a>
 ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>assume </b><a NAME="E2:88_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="rcomp_1.html#V3">open</a>
 ;<br/><a NAME="E3:88_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="rcomp_1.html#V2">closed</a>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D5">RCOMP_1:def 5</a></i>;<br/><a NAME="E4:88_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:88_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="pscomp_1.html#D25" target="_self">Def25</a></i>;<br/><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> = 
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <a href="numbers.html#K1">REAL</a> </span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T138">FUNCT_1:138</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T48">FUNCT_2:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>
;<br/><a NAME="E6:88_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E4:88_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/><b>hence </b><a NAME="E7:88_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#T22">PRE_TOPC:22</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:88"/><i><font color="Green">E57</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#V3">open</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#V3">open</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</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" target="_self">PSCOMP_1:def 25</a><br/>

<b>assume </b><a NAME="E3:88"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="rcomp_1.html#V2">closed</a>
 ;<br/>

<a NAME="E4:88"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> 
 
;<br/>
<a NAME="E5:88"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="rcomp_1.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E3:88"><i><font color="Green">E58</font></i></a>, <a class="ref" href="rcomp_1.html#D5">RCOMP_1:def 5</a></i>;<br/>
<a NAME="E6:88"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E2:88"><i><font color="Green">E57</font></i></a></i>;<br/>
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> = 
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <a href="numbers.html#K1">REAL</a> </span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T138">FUNCT_1:138</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T48">FUNCT_2:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E8:88"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <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="pscomp_1.html#E6:88"><i><font color="Green">E59</font></i></a>, <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/>


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