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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/>

<b>assume </b><a NAME="E1:32"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="t_0topsp.html#R1">are_homeomorphic</a>  or ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>(  <a href="relset_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> ) )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1"/>
( <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="t_0topsp.html#R1">are_homeomorphic</a>  or ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>(  <a href="relset_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="yellow14.html#E1:32"><i><font color="Green">E9</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1"><b>suppose </b></a><a NAME="E1:32_1_1"/>
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="t_0topsp.html#R1">are_homeomorphic</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:32_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i><a class="ref" href="t_0topsp.html#D1">T_0TOPSP:def 1</a></i>;<br/><a NAME="E4:32_1_1"/>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="yellow14.html#E3:32_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><b>hence </b><a NAME="E5:32_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a> iff <font color="Maroon">c<sub>2</sub></font> is <a href="struct_0.html#V3">empty</a> )
 <b>by </b><i><a class="ref" href="yellow14.html#T4" target="_self">Th4</a>, <a class="ref" href="yellow14.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2"><b>suppose </b></a><a NAME="E1:32_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>(  <a href="relset_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_1_2"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a> iff <font color="Maroon">c<sub>2</sub></font> is <a href="struct_0.html#V3">empty</a> )
 <b>by </b><i><a class="ref" href="yellow14.html#T4" target="_self">Th4</a>, <a class="ref" href="yellow14.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
