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<span class="kw">let </span><font color="Maroon" title="c2">X0</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">X0</font> is  <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>  implies  not <font color="Maroon" title="c2">X0</font> is  <a href="tex_3.html#V3" title="TEX_3:attr.3">boundary</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_1_1"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c2">X0</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"> not <font color="Maroon" title="c2">X0</font> is  <a href="tex_3.html#V3" title="TEX_3:attr.3">boundary</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c3">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font> as   <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="E2:39_1_1_1"/>
<font color="Maroon" title="c2">X0</font> is  <a href="tex_3.html#V3" title="TEX_3:attr.3">boundary</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:39_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">A</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tex_3.html#D3" target="_self" title="TEX_3:def.3">Def3</a></span>;<br/><a NAME="E4:39_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A2</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T84" title="TOPS_1:th.84">TOPS_1:84</a></span>;<br/><a NAME="E5:39_1_1_1"/>
<font color="Maroon" title="c3">A</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="txt" href="#E1:39_1_1"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="tsep_1.html#T16" title="TSEP_1:th.16">TSEP_1:16</a></span>;<br/><span class="kw">hence </span><a NAME="E6:39_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_1_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:39_1_1"/>
not <font color="Maroon" title="c2">X0</font> is  <a href="tex_3.html#V3" title="TEX_3:attr.3">boundary</a> 
 ; <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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