<?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>, <font color="Maroon">V</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">P1</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">P2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font></span>)</span>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P2</font></span>)</span>,<font color="Maroon">V</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:50"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P2</font>
 <b>and </b><br/><a NAME="E2:50"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E3:50"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">P3</font> = <font color="Maroon">P2</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="jordan6.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P3</font></span>)</span>,<font color="Maroon">V</font> ;<br/>
<a NAME="E6:50"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E7:50"/><b>then </b><i><font color="Green">E51</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font>
 
;<br/>
<a NAME="E8:50"/><i><font color="Green">E52</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E9:50"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P3</font></span>)</span> <b>by </b><i><a class="ref" href="jordan6.html#T2" target="_self">Th2</a>, , <a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<a NAME="E11:50"/>
<font color="Maroon">f1</font> is <a href="pre_topc.html#V5">continuous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<a NAME="E1:50_1"/>
for <font color="Olive">F1</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P2</font></span>)</span>  st <font color="Olive">F1</font> is <a href="pre_topc.html#V4">closed</a> holds <br/><font color="Maroon">f1</font> <a href="relset_1.html#K11">"</a> <font color="Olive">F1</font> is <a href="pre_topc.html#V4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">F1</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P2</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:50_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">F1</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">P1'</font> = <font color="Maroon">P1</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P3</font></span>)</span> <b>by </b><i><a class="ref" href="jordan6.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E3:50_1_1"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E4:50_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font>
 
;<br/>
<b>reconsider </b><font color="Maroon">P4</font> = <font color="Maroon">F1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P1'</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P3</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1'</font></span>)</span> <b>by </b><i><a class="ref" href="tops_2.html#T38">TOPS_2:38</a></i>;<br/>
<a NAME="E6:50_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">S</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P3</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1'</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E7:50_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">P4</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E58</font></i>: <font color="Maroon">f1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#T48">FUNCT_2:48</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E9:50_1_1"/>
<font color="Maroon">f1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">F1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T167">RELAT_1:167</a></i>;<br/>
<b>then </b><font color="Maroon">f1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">F1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">F1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P4</font>
<b>by </b><i><a class="ref" href="funct_1.html#T137">FUNCT_1:137</a></i>
;<br/>
<b>hence </b><a NAME="E11:50_1_1"/>
<font color="Maroon">f1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">F1</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="jordan6.html#E1:50"><i><font color="Green">E37</font></i></a>, <a class="ref" href="jordan6.html#T4" target="_self">Th4</a>, , <a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:50_1"/>
<font color="Maroon">f1</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:50"/><b>then </b>
<font color="Maroon">g1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="jordan6.html#T3" target="_self">Th3</a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<b>hence </b><a NAME="E13:50"/>
 ex <font color="Olive">h</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<font color="Maroon">V</font> st <br/>( <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> &amp; <font color="Olive">h</font> is <a href="pre_topc.html#V5">continuous</a> )
 ;<br/>


</div>
