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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">V130</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/><span class="kw">let </span><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">V130</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>



<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:41"/>
( <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E3:41"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A1</font></span>: 
( <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>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T16" title="TSEP_1:th.16">TSEP_1:16</a></span>;<br/>
<a NAME="E4:41"/>
( <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>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tsp_1.html#T15" target="_self" title="TSP_1:th.15">Th15</a></span>;<br/>
<a NAME="E5:41"/><span class="kw">then </span>
( 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> &amp; <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>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="tsp_1.html#T9" target="_self" title="TSP_1:th.9">Th9</a>, <a class="ref" href="tsep_1.html#D2" title="TSEP_1:def.2">TSEP_1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:41"/>
<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 class="kw">by</span> <span class="lab"><a class="ref" href="tsp_1.html#T15" target="_self" title="TSP_1:th.15">Th15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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