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<b>let </b><font color="Maroon">X0</font> be  non <a href="struct_0.html#V3">empty</a> <a href="tex_2.html#V2" target="_self">proper</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/>

<b>reconsider </b><font color="Maroon">A</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">B</font> = <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> ;<br/>
<a NAME="E2:60_1_1"/>
<font color="Maroon">A</font> is <a href="tex_2.html#V1" target="_self">proper</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:60_1_1"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:60_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
( <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="tops_3.html#T1">TOPS_3:1</a>, <a class="ref" href="tops_3.html#T2">TOPS_3:2</a></i>;<br/>
<b>assume </b><a NAME="E5:60_1_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">X0</font> is <a href="tsep_1.html#V1">open</a> or <font color="Maroon">X0</font> is <a href="borsuk_1.html#V1">closed</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_1_1_1_1"/>
( <font color="Maroon">X0</font> is <a href="tsep_1.html#V1">open</a> or <font color="Maroon">X0</font> is <a href="borsuk_1.html#V1">closed</a> )
 </span><b>by </b><i><a class="ref" href="tex_2.html#T1" target="_self">Th1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:60_1_1_1_1_1"/>
<font color="Maroon">X0</font> is <a href="tsep_1.html#V1">open</a>
 ;<br/><div class="add"><a NAME="E2:60_1_1_1_1_1"/><b>then </b>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tsep_1.html#D1">TSEP_1:def 1</a></i>;<br/><a NAME="E3:60_1_1_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E4:60_1_1_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tdlat_3.html#D2">TDLAT_3:def 2</a></i>;<br/><b>hence </b><a NAME="E5:60_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tex_2.html#E3:60_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:60_1_1_1_1_2"/>
<font color="Maroon">X0</font> is <a href="borsuk_1.html#V1">closed</a>
 ;<br/><div class="add"><a NAME="E2:60_1_1_1_1_2"/><b>then </b>
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D14">BORSUK_1:def 14</a></i>;<br/><a NAME="E3:60_1_1_1_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/><a NAME="E4:60_1_1_1_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E5:60_1_1_1_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tdlat_3.html#D2">TDLAT_3:def 2</a></i>;<br/><b>hence </b><a NAME="E6:60_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:60_1_1"/>
contradiction
 ;<br/>


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