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<b>let </b><font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c4">B</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">S</font>;<br/>





<b>assume </b><a NAME="E1:6"/><i><font color="Green" title="E6">A1</font></i>: 
( <font color="Maroon" title="c3">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c4">B</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> )
 ;<br/>

<a NAME="E2:6"/>
for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="weierstr.html#K1" title="WEIERSTR:func.1">"</a> <font color="Maroon" title="c4">B</font> holds <br/><font color="Olive" title="b1">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><a NAME="E1:6_1"/><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="weierstr.html#K1" title="WEIERSTR:func.1">"</a> <font color="Maroon" title="c4">B</font>
 ;<br/>

<b>thus </b><a NAME="E2:6_1"/>
<font color="Maroon" title="c5">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c6">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E2:6_1_1"/><i><font color="Green" title="E8">A3</font></i>: 
( <font color="Maroon" title="c6">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c6">C</font> )
 <b>by </b><i><a class="txt" href="weierstr.html#E1:6_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="weierstr.html#D1" target="_self" title="WEIERSTR:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">C</font> = <font color="Maroon" title="c6">C</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<a NAME="E4:6_1_1"/><i><font color="Green" title="E9">A4</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E5:6_1_1"/>
<font color="Maroon" title="c7">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="weierstr.html#E1:6"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="weierstr.html#E2:6_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:6_1_1"/>
<font color="Maroon" title="c5">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="weierstr.html#E1:6"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="weierstr.html#E2:6_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="weierstr.html#E4:6_1_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6"/>
<font color="Maroon" title="c3">f</font> <a href="weierstr.html#K1" title="WEIERSTR:func.1">"</a> <font color="Maroon" title="c4">B</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 <b>by </b><i><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></i>;<br/>


</div>
