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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <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> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tsep_2.html#R2">constitute_a_decomposition</a> 
 ;<br/>

<a NAME="E2:1"/><b>then </b><i><font color="Green">E2</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i><a class="ref" href="tsep_2.html#T4">TSEP_2:4</a></i>;<br/>
<b>thus </b><a NAME="E3:1"/>
( not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> implies <font color="Maroon">c<sub>3</sub></font> is <a href="tex_2.html#V1">proper</a> )
 <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">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<a NAME="E2:1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="tex_3.html#E2:1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="tops_3.html#T1">TOPS_3:1</a></i>;<br/>
<b>hence </b><a NAME="E3:1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="tex_2.html#V1">proper</a>
 <b>by </b><i><a class="ref" href="tex_2.html#T5">TEX_2:5</a></i>;<br/>


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

<b>assume </b><a NAME="E4:1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="tex_2.html#V1">proper</a>
 ;<br/>

<a NAME="E5:1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tex_2.html#T5">TEX_2:5</a></i>;<br/>
<b>hence </b><a NAME="E6:1"/>
not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="tex_3.html#E2:1"><i><font color="Green">E2</font></i></a></i>;<br/>


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