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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#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> &amp; <font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a>  holds <br/><font color="Olive" title="b1">X1</font> is  <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font></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#M1" title="PRE_TOPC:mode.1">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="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> &amp; <font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a>  implies <font color="Maroon" title="c2">X1</font> is  <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> )</span><br/>





<b>assume </b><a NAME="E1:81"/><i><font color="Green" title="E55">A1</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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>
 ; <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">X1</font>,<font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a>  or <font color="Maroon" title="c2">X1</font> is  <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> )</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> as   <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" target="_self" title="TSEP_1:th.1">Th1</a></i>;<br/>
<b>reconsider </b><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   <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" target="_self" title="TSEP_1:th.1">Th1</a></i>;<br/>
<a NAME="E4:81"/><i><font color="Green" title="E56">A2</font></i>: 
<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="#E1:81"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="tsep_1.html#D2" target="_self" title="TSEP_1:def.2">Def2</a></i>;<br/>
<b>assume </b><a NAME="E5:81"/>
<font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c3">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</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> is  <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E6:81"/><b>then </b>
<font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c5">A2</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 
<b>by </b><i><a class="ref" href="tsep_1.html#D8" target="_self" title="TSEP_1:def.8">Def8</a></i>;<br/>
<a NAME="E7:81"/><b>then </b>
( <font color="Maroon" title="c4">A1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c5">A2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 
<b>by </b><i><a class="txt" href="#E4:81"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="connsp_1.html#T5" title="CONNSP_1:th.5">CONNSP_1:5</a></i>;<br/>
<b>hence </b><a NAME="E8:81"/>
<font color="Maroon" title="c2">X1</font> is  <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T11" target="_self" title="TSEP_1:th.11">Th11</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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