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

<b>let </b><font color="Maroon">T</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:6_1"/>
<font color="Maroon">A</font> <a href="connsp_1.html#R4">is_a_component_of</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font>
 ;<br/><a NAME="E2:6_1"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="sprect_1.html#T7">SPRECT_1:7</a></i>;<br/><b>hence </b><a NAME="E3:6_1"/>
<font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:6"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E4:6"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:6"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font></span>)</span></span>)</span>  st <font color="Olive">C</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Olive">C</font> holds <br/><font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:6_2"/>
( <font color="Maroon">C</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> )
 ;<br/>

<a NAME="E2:6_2"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:6_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:6"/><b>then </b>
<font color="Maroon">B</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="connsp_1.html#D5">CONNSP_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:6"/>
<font color="Maroon">A</font> <a href="connsp_1.html#R4">is_a_component_of</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a></i>;<br/>


</div>
