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<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">X1</font>, <font color="Olive" title="b2">X2</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#V6" title="PRE_TOPC:attr.6">V115</a>  <a href="tsp_1.html#M2" title="TSP_1:mode.2">SubSpace</a> of <font color="Maroon" title="c1">X</font>  st ( <font color="Olive" title="b1">X1</font> is <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a> or <font color="Olive" title="b2">X2</font> is <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a> ) holds <br/><font color="Olive" title="b1">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Olive" title="b2">X2</font> is <a href="pre_topc.html#V6" title="PRE_TOPC:attr.6">T_0</a></span><br/><b>let </b><font color="Maroon" title="c2">X1</font>, <font color="Maroon" title="c3">X2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#V6" title="PRE_TOPC:attr.6">V115</a>  <a href="tsp_1.html#M2" title="TSP_1:mode.2">SubSpace</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">X1</font> is <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a> or <font color="Maroon" title="c3">X2</font> is <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a> ) implies <font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c3">X2</font> is <a href="pre_topc.html#V6" title="PRE_TOPC:attr.6">T_0</a> )</span><br/>



<b>reconsider </b><font color="Maroon" title="c4">A1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X1</font>, <font color="Maroon" title="c5">A2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X2</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></i>;<br/>
<b>assume </b><a NAME="E2:39"/>
( <font color="Maroon" title="c2">X1</font> is <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a> or <font color="Maroon" title="c3">X2</font> is <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c3">X2</font> is <a href="pre_topc.html#V6" title="PRE_TOPC:attr.6">T_0</a></span><br/>

<a NAME="E3:39"/><b>then </b><i><font color="Green" title="E25">A1</font></i>: 
( <font color="Maroon" title="c4">A1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or <font color="Maroon" title="c5">A2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="ref" href="tsep_1.html#T16" title="TSEP_1:th.16">TSEP_1:16</a></i>;<br/>
<a NAME="E4:39"/><i><font color="Green" title="E26">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c3">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">A2</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#D2" title="TSEP_1:def.2">TSEP_1:def 2</a></i>;<br/>
<a NAME="E5:39"/>
( <font color="Maroon" title="c4">A1</font> is <a href="tsp_1.html#V1" title="TSP_1:attr.1">T_0</a> &amp; <font color="Maroon" title="c5">A2</font> is <a href="tsp_1.html#V1" title="TSP_1:attr.1">T_0</a> )
 
<b>by </b><i><a class="ref" href="tsp_1.html#T15" target="_self" title="TSP_1:th.15">Th15</a></i>;<br/>
<a NAME="E6:39"/><b>then </b>
<font color="Maroon" title="c4">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">A2</font> is <a href="tsp_1.html#V1" title="TSP_1:attr.1">T_0</a>
 
<b>by </b><i><a class="txt" href="#E3:39"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="tsp_1.html#T9" target="_self" title="TSP_1:th.9">Th9</a></i>;<br/>
<b>hence </b><a NAME="E7:39"/>
<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c3">X2</font> is <a href="pre_topc.html#V6" title="PRE_TOPC:attr.6">T_0</a>
 <b>by </b><i><a class="txt" href="#E4:39"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="tsp_1.html#T15" target="_self" title="TSP_1:th.15">Th15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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