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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  holds <br/>( not <font color="Olive" title="b1">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff <font color="Olive" title="b2">B</font> is <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a> )</span><br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  implies ( not <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff <font color="Maroon" title="c3">B</font> is <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a> ) )</span><br/>





<b>assume </b><a NAME="E1:1"/>
<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff <font color="Maroon" title="c3">B</font> is <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a> )</span><br/>

<a NAME="E2:1"/><b>then </b><i><font color="Green" title="E2">A1</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
<b>by </b><i><a class="ref" href="tsep_2.html#T4" title="TSEP_2:th.4">TSEP_2:4</a></i>;<br/>
<b>thus </b><a NAME="E3:1"/>
( not <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies <font color="Maroon" title="c3">B</font> is <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">B</font> is <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a> implies  not <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1"/>
not <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">B</font> is <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a></span><br/>

<a NAME="E2:1_1"/><b>then </b>
<font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="#E2:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="tops_3.html#T1" title="TOPS_3:th.1">TOPS_3:1</a></i>;<br/>
<b>hence </b><a NAME="E3:1_1"/>
<font color="Maroon" title="c3">B</font> is <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>
 <b>by </b><i><a class="ref" href="subset_1.html#D7" title="SUBSET_1:def.7">SUBSET_1:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E4:1"/>
<font color="Maroon" title="c3">B</font> is <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a></span><br/>

<b>hence </b><a NAME="E5:1"/>
not <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="#E2:1"><i><font color="Green" title="E2">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
