<?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>, <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/>



<a NAME="E1:15"/><i><font color="Green">E27</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/><b>thus </b><a NAME="E2:15_1"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>  st <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> holds <br/> ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/>



<b>assume </b><a NAME="E1:15_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>

<a NAME="E2:15_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</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="E3:15_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E4:15_1_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T145">FUNCT_1:145</a></i>;<br/>
<b>hence </b><a NAME="E5:15_1_1"/>
 ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 <b>by </b><i>, </i>;<br/>


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

<b>assume </b><a NAME="E3:15"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>  st <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> holds <br/> ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> )
 ;<br/>

<a NAME="E4:15"/>
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">f</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="15_2"><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="E1:15_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:15_2"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> iff  ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">Q</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">Q</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_2_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font>
 ;<br/><a NAME="E2:15_2_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:15_2_1_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E6:15_2_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T178">RELAT_1:178</a></i>;<br/><a NAME="E7:15_2_1_1"/>
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T146">FUNCT_1:146</a></i>;<br/><a NAME="E8:15_2_1_1"/><b>then </b>
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E9:15_2_1_1"/>
 ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">Q</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">Q</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:15_2_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> iff  ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">Q</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">Q</font> ) )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:15_2"/>
<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_1.html#T57">TOPS_1:57</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:15"/>
<font color="Maroon">f</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>
