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

<b>thus </b><a NAME="E1:10_1_1"/>
( <font color="Maroon">f</font> is <a href="pscomp_1.html#V9">continuous</a> implies for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font><br/> for <font color="Olive">N</font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font>  ex <font color="Olive">V</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">p</font> st <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">V</font> <a href="tarski.html#R1">c=</a> <font color="Olive">N</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">f</font> is <a href="pscomp_1.html#V9">continuous</a>
 ;<br/>

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">N</font> be    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>;<br/>



<a NAME="E2:10_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">N</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="jct_misc.html#T5">JCT_MISC:5</a></i>;<br/>
<a NAME="E3:10_1_1_1"/>
<font color="Maroon">N</font> <a href="subset_1.html#K3">`</a>  is <a href="rcomp_1.html#V2">closed</a>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D5">RCOMP_1:def 5</a></i>;<br/>
<a NAME="E4:10_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i>, <a class="ref" href="pscomp_1.html#D25">PSCOMP_1:def 25</a></i>;<br/>
<a NAME="E5:10_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">N</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/>
<a NAME="E6:10_1_1_1"/>
<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">N</font>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T37">RCOMP_1:37</a></i>;<br/>
<a NAME="E7:10_1_1_1"/><b>then </b>
<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">N</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">V</font> = <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">N</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p</font> <b>by </b><i>, <a class="ref" href="connsp_2.html#T5">CONNSP_2:5</a></i>;<br/>
<b>take </b>
<font color="Maroon">V</font>
;<br/>

<b>thus </b><a NAME="E9:10_1_1_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T145">FUNCT_1:145</a></i>;<br/>


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

<b>assume </b><a NAME="E2:10_1_1"/><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">T</font><br/> for <font color="Olive">N</font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font>  ex <font color="Olive">V</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">p</font> st <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">V</font> <a href="tarski.html#R1">c=</a> <font color="Olive">N</font>
 ;<br/>

<b>let </b><font color="Maroon">Y</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pscomp_1.html#D25">PSCOMP_1:def 25</a><br/>

<b>assume </b><a NAME="E3:10_1_1"/>
<font color="Maroon">Y</font> is <a href="rcomp_1.html#V2">closed</a>
 ;<br/>

<a NAME="E4:10_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="rcomp_1.html#V2">closed</a>
 
;<br/>
<a NAME="E5:10_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">Y</font> <a href="subset_1.html#K3">`</a>  is <a href="rcomp_1.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D5">RCOMP_1:def 5</a></i>;<br/>
<a NAME="E6:10_1_1"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K3">`</a>  holds <br/> ex <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">A</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">p</font> &amp; <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K3">`</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_2"><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">T</font>;<br/>

<b>assume </b><a NAME="E1:10_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<a NAME="E2:10_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="jct_misc.html#T5">JCT_MISC:5</a></i>;<br/>
<a NAME="E3:10_1_1_2"/><b>then </b>
<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">Y</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<b>then consider </b><font color="Maroon">N</font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font><b> such that </b><br/><a NAME="E5:10_1_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T39">RCOMP_1:39</a></i>;<br/>
<b>consider </b><font color="Maroon">V</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p</font><b> such that </b><br/><a NAME="E7:10_1_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">V</font>
;<br/>

<b>thus </b><a NAME="E8:10_1_1_2"/>
<font color="Maroon">V</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p</font>
 ;<br/>

<a NAME="E9:10_1_1_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:10_1_1_2"/><b>then </b><i><font color="Green">E56</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">V</font></span>)</span> <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">Y</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T178">RELAT_1:178</a></i>;<br/>
<a NAME="E11:10_1_1_2"/>
<font color="Maroon">V</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">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T50">FUNCT_2:50</a></i>;<br/>
<a NAME="E12:10_1_1_2"/><b>then </b>
<font color="Maroon">V</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">Y</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E13:10_1_1_2"/>
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="jct_misc.html#T5">JCT_MISC:5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:10_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T9">CONNSP_2:9</a></i>;<br/>
<a NAME="E8:10_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/>
<b>hence </b><a NAME="E9:10_1_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>


</div>
