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

<b>let </b><font color="Maroon">S1</font> be   <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">T1</font> be   <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">S2</font> be   <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">T2</font> be   <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">S1</font>,<font color="Maroon">S2</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T1</font>,<font color="Maroon">T2</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:46"/><i><font color="Green">E34</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S1</font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T1</font> #)
 <b>and </b><br/><a NAME="E2:46"/><i><font color="Green">E35</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S2</font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T2</font> #)
 <b>and </b><br/><a NAME="E3:46"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font>;<br/><b>assume </b><a NAME="E1:46_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">P2</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/><b>reconsider </b><font color="Maroon">P1</font> = <font color="Maroon">P2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S2</font> <b>by </b><i/>;<br/><a NAME="E3:46_1"/>
<font color="Maroon">P1</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, , <a class="ref" href="tops_3.html#T79">TOPS_3:79</a></i>;<br/><a NAME="E4:46_1"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/><b>hence </b><a NAME="E5:46_1"/>
<font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P2</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, , <a class="ref" href="tops_3.html#T79">TOPS_3:79</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:46"/>
<font color="Maroon">g</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>
