<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">K0</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K0</font></span>)</span>,<font color="Maroon">Y</font>;<br/>









<b>assume </b><a NAME="E1:10"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <font color="Maroon">K0</font> )
 ;<br/>

<a NAME="E2:10"/>
for <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>  st <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/>

<b>assume </b><a NAME="E1:10_1"/>
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:10_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<a NAME="E3:10_1"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">K0</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E4:10_1"/><b>then </b>
<font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K0</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T139">FUNCT_1:139</a></i>;<br/>
<b>hence </b><a NAME="E5:10_1"/>
<font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10"/>
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>


</div>
