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

<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">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">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">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/>







<b>thus </b><a NAME="E1:52"/>
( <font color="Maroon">f</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font> implies 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; <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="Olive">G</font> holds <br/> ex <font color="Olive">H</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">H</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">H</font> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">H</font> <a href="tarski.html#R1">c=</a> <font color="Olive">G</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_1"/><i><font color="Green">E23</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/>

<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="E2:52_1"/>
( <font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <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> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">G0</font> = <font color="Maroon">G</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <b>by </b><i><a class="ref" href="connsp_2.html#T5">CONNSP_2:5</a></i>;<br/>
<b>consider </b><font color="Maroon">H0</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font><b> such that </b><br/><a NAME="E5:52_1"/><i><font color="Green">E29</font></i>: 
<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">G0</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:52"><i><font color="Green">E39</font></i></a>, <a class="txt" href="tmap_1.html#E8:52_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>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="E7:52_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">H</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E8:52_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H0</font>
 <b>and </b><br/><a NAME="E9:52_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#T8">CONNSP_2:8</a></i>;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

<a NAME="E10:52_1"/>
<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">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H0</font>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E4:52"><i><font color="Green">E42</font></i></a>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<b>hence </b><a NAME="E11:52_1"/>
( <font color="Maroon">H</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> &amp; <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#E5:52"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


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

<b>assume </b><a NAME="E2:52"/><i><font color="Green">E39</font></i>: 
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; <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="Olive">G</font> holds <br/> ex <font color="Olive">H</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">H</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">H</font> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">H</font> <a href="tarski.html#R1">c=</a> <font color="Olive">G</font> )
 ;<br/>

<b>let </b><font color="Maroon">G0</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tmap_1.html#D2" target="_self">TMAP_1:def 2</a><br/>

<b>consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E4:52"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E5:52"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G0</font>
 <b>and </b><br/><a NAME="E6:52"/><i><font color="Green">E44</font></i>: 
<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="ref" href="connsp_2.html#T8">CONNSP_2:8</a></i>;<br/>
<b>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="E8:52"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">H</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E9:52"/><i><font color="Green">E46</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="E10:52"/><i><font color="Green">E47</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#E2:52"><i><font color="Green">E39</font></i></a>, , <a class="txt" href="tmap_1.html#E4:52"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">H0</font> = <font color="Maroon">H</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font> <b>by </b><i><a class="txt" href="tmap_1.html#E5:52"><i><font color="Green">E43</font></i></a>, <a class="txt" href="tmap_1.html#E6:52"><i><font color="Green">E44</font></i></a>, <a class="ref" href="connsp_2.html#T5">CONNSP_2:5</a></i>;<br/>
<b>take </b>
<font color="Maroon">H0</font>
;<br/>

<b>thus </b><a NAME="E12:52"/>
<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">G0</font>
 <b>by </b><i>, <a class="txt" href="tmap_1.html#E8:52"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div>
