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

<b>let </b><font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<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">X0</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">x0</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X0</font>;<br/>











<b>assume </b><a NAME="E1:65"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 ;<br/>

<b>assume </b><a NAME="E2:65"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">f</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E3:65"/>
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> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X0</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">G</font> holds <br/> ex <font color="Olive">H0</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X0</font> st <br/>( <font color="Olive">H0</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">H0</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X0</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Olive">H0</font> <a href="tarski.html#R1">c=</a> <font color="Olive">G</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_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><b>that </b><br/><a NAME="E1:65_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:65_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X0</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E3:65_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 
<b>by </b><i>, , <a class="txt" href="tmap_1.html#E11:65_1"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">H</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:65_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">H</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E6:65_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E7:65_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="txt" href="tmap_1.html#E5:65_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">C</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">H0</font> = <font color="Maroon">H</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X0</font> <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>take </b>
<font color="Maroon">H0</font>
;<br/>

<a NAME="E10:65_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">H0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X0</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H0</font> )
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E11:65_1"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X0</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H0</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T161">RELAT_1:161</a></i>;<br/>
<a NAME="E12:65_1"/>
( <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H0</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font></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#K10">.:</a> <font color="Maroon">C</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font></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#K10">.:</a> <font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T154">RELAT_1:154</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E13:65_1"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E14:65_1"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E15:65_1"/>
( <font color="Maroon">H0</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H0</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X0</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> )
 <b>by </b><i>, , , , <a class="ref" href="tops_2.html#T32">TOPS_2:32</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:65"/>
<font color="Maroon">f</font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X0</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x0</font>
 <b>by </b><i/>;<br/>


</div>
