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<b>let </b><font color="Maroon" title="c1">T</font> be    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:86_1"/><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a>
 ;<br/><b>let </b><font color="Maroon" title="c3">Y</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>assume </b><a NAME="E2:86_1"/>
<font color="Maroon" title="c3">Y</font> is <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>
 ;<br/><a NAME="E3:86_1"/><b>then </b>
<font color="Maroon" title="c3">Y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D5" title="RCOMP_1:def.5">RCOMP_1:def 5</a></i>;<br/><a NAME="E4:86_1"/><b>then </b><i><font color="Green" title="E55">A2</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:86_1"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="pscomp_1.html#D25" target="_self" title="PSCOMP_1:def.25">Def25</a></i>;<br/><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T138" title="FUNCT_1:th.138">FUNCT_1:138</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T48" title="FUNCT_2:th.48">FUNCT_2:48</a></i>
;<br/><a NAME="E6:86_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E4:86_1"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="pre_topc.html#D6" title="PRE_TOPC:def.6">PRE_TOPC:def 6</a></i>;<br/><b>hence </b><a NAME="E7:86_1"/>
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">Y</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#T22" title="PRE_TOPC:th.22">PRE_TOPC:22</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:86"/><i><font color="Green" title="E54">A3</font></i>: 
for <font color="Olive" title="b1">Y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">Y</font> is <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> holds <br/><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">Y</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">Y</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pscomp_1.html#D25" target="_self" title="PSCOMP_1:def.25">PSCOMP_1:def 25</a><br/>

<b>assume </b><a NAME="E3:86"/><i><font color="Green" title="E55">A4</font></i>: 
<font color="Maroon" title="c3">Y</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>
 ;<br/>

<a NAME="E4:86"/>
<font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
;<br/>
<a NAME="E5:86"/><b>then </b>
<font color="Maroon" title="c3">Y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E3:86"><i><font color="Green" title="E55">A4</font></i></a>, <a class="ref" href="rcomp_1.html#D5" title="RCOMP_1:def.5">RCOMP_1:def 5</a></i>;<br/>
<a NAME="E6:86"/><b>then </b><i><font color="Green" title="E56">A5</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E2:86"><i><font color="Green" title="E54">A3</font></i></a></i>;<br/>
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T138" title="FUNCT_1:th.138">FUNCT_1:138</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T48" title="FUNCT_2:th.48">FUNCT_2:48</a></i>
;<br/>
<b>hence </b><a NAME="E8:86"/>
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">Y</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E6:86"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="pre_topc.html#D6" title="PRE_TOPC:def.6">PRE_TOPC:def 6</a></i>;<br/>


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