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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>thus </b><a NAME="E1:54"/>
( ( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="connsp_1.html#R1">are_separated</a>  &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="connsp_1.html#R1">are_separated</a>  ) iff <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="connsp_1.html#R1">are_separated</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:54_1"/>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="connsp_1.html#R1">are_separated</a>  &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="connsp_1.html#R1">are_separated</a>  implies <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="connsp_1.html#R1">are_separated</a>  )
 <b>by </b><i><a class="ref" href="connsp_1.html#T9">CONNSP_1:9</a></i>;<br/>

<b>thus </b><a NAME="E2:54_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="connsp_1.html#R1">are_separated</a>  implies ( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="connsp_1.html#R1">are_separated</a>  &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="connsp_1.html#R1">are_separated</a>  ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="connsp_1.html#R1">are_separated</a> 
 ;<br/>

<a NAME="E2:54_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E3:54_1_1"/>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="connsp_1.html#R1">are_separated</a>  &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="connsp_1.html#R1">are_separated</a>  )
 <b>by </b><i><a class="txt" href="tsep_1.html#E1:54_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="connsp_1.html#T8">CONNSP_1:8</a></i>;<br/>


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


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


</div>
