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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">S</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">T</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>





<b>assume </b><a NAME="E1:16"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">P</font> is <a href="compts_1.html#V6">compact</a> &amp; <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a> )
 ;<br/>

<a NAME="E2:16"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 
;<br/>
<a NAME="E3:16"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:16"/>
for <font color="Olive">F0</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">S</font>  st <font color="Olive">F0</font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> &amp; <font color="Olive">F0</font> is <a href="tops_2.html#V1">open</a> holds <br/> ex <font color="Olive">G</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Olive">F0</font> &amp; <font color="Olive">G</font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> &amp; <font color="Olive">G</font> is <a href="finset_1.html#V1">finite</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">F0</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">F0</font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> &amp; <font color="Maroon">F0</font> is <a href="tops_2.html#V1">open</a> )
 ;<br/>

<a NAME="E2:16_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">F0</font>
 
<b>by </b><i><a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">B0</font> = <font color="Maroon">f</font> <a href="weierstr.html#K1" target="_self">"</a> <font color="Maroon">F0</font>;<br/>
<a NAME="E3:16_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> <a href="weierstr.html#K1" target="_self">"</a> <font color="Maroon">F0</font> is <a href="tops_2.html#V1">open</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E4:16_1"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="weierstr.html#K1" target="_self">"</a> <font color="Maroon">F0</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>thus </b><a NAME="E1:16_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="weierstr.html#K1" target="_self">"</a> <font color="Maroon">F0</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:16_1_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E4:16_1_1_1"/><i><font color="Green">E38</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>  <a href="tarski.html#K3">union</a> <font color="Maroon">F0</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:16_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></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"><a href="tarski.html#K3">union</a> <font color="Maroon">F0</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:16_1_1_1_1"/><b>then </b>
( <font color="Maroon">y</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">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E3:16_1_1_1_1"/><b>then </b>
( <font color="Maroon">y</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">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">F0</font> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:16_1_1_1_1"/>
<font color="Maroon">y</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"><a href="tarski.html#K3">union</a> <font color="Maroon">F0</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:16_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">F0</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="weierstr.html#K1" target="_self">"</a> <font color="Maroon">F0</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_1_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">y</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"><a href="tarski.html#K3">union</a> <font color="Maroon">F0</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E2:16_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="weierstr.html#K1" target="_self">"</a> <font color="Maroon">F0</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_1_1_1_2_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">y</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">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">F0</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">Q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:16_1_1_1_2_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F0</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:16_1_1_1_2_1"/>
 ex <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> &amp; <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="weierstr.html#K1" target="_self">"</a> <font color="Maroon">F0</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">Z</font> = <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Q</font>;<br/>
<b>take </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Q</font>
;<br/>

<b>thus </b><a NAME="E1:16_1_1_1_2_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="weierstr.html#K1" target="_self">"</a> <font color="Maroon">F0</font> )
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:16_1_1_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="weierstr.html#K1" target="_self">"</a> <font color="Maroon">F0</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E7:16_1_1_1"/>
( <font color="Maroon">x</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_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:16_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E9:16_1_1_1"/><b>then </b>
<font color="Maroon">x</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"><a href="tarski.html#K3">union</a> <font color="Maroon">F0</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:16_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="weierstr.html#K1" target="_self">"</a> <font color="Maroon">F0</font></span>)</span>
 <b>by </b><i/>;<br/>


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


</div><b>end;</b></div>
<a NAME="E5:16_1"/><b>then </b>
<font color="Maroon">f</font> <a href="weierstr.html#K1" target="_self">"</a> <font color="Maroon">F0</font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">B</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E7:16_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="weierstr.html#K1" target="_self">"</a> <font color="Maroon">F0</font> &amp; <font color="Maroon">B</font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">P</font> &amp; <font color="Maroon">B</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i>, , <a class="ref" href="compts_1.html#D7">COMPTS_1:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E9:16_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F0</font> &amp; <font color="Maroon">G</font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> &amp; <font color="Maroon">G</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<b>thus </b><a NAME="E10:16_1"/>
( <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F0</font> &amp; <font color="Maroon">G</font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> &amp; <font color="Maroon">G</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:16"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i><a class="ref" href="compts_1.html#D7">COMPTS_1:def 7</a></i>;<br/>


</div>
