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<b>let </b><font color="Maroon" title="c1">F</font>, <font color="Maroon" title="c2">G</font> be  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a> <a href="borsuk_5.html#V1" title="BORSUK_5:attr.1">with_proper_subsets</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:58"/><i><font color="Green" title="E45">A1</font></i>: 
( <font color="Maroon" title="c1">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> &amp; <font color="Maroon" title="c2">G</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> )
 ;<br/>

<b>assume </b><a NAME="E2:58"/>
<font color="Maroon" title="c1">F</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">G</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:58"/><i><font color="Green" title="E46">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font>
 <b>and </b><br/><a NAME="E5:58"/><i><font color="Green" title="E47">A3</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">x</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="txt" href="kurato_1.html#E4:58"><i><font color="Green" title="E46">A2</font></i></a></i>;<br/>
<a NAME="E7:58"/><i><font color="Green" title="E48">A4</font></i>: 
<font color="Maroon" title="c4">x</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="kurato_1.html#E1:58"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="kurato_1.html#E4:58"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></i>;<br/>
<a NAME="E8:58"/>
<font color="Maroon" title="c4">x</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="kurato_1.html#E1:58"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="kurato_1.html#E5:58"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="tops_2.html#D2" title="TOPS_2:def.2">TOPS_2:def 2</a></i>;<br/>
<a NAME="E9:58"/><b>then </b>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 
<b>by </b><i><a class="txt" href="kurato_1.html#E7:58"><i><font color="Green" title="E48">A4</font></i></a>, <a class="ref" href="borsuk_5.html#T57" title="BORSUK_5:th.57">BORSUK_5:57</a></i>;<br/>
<b>hence </b><a NAME="E10:58"/>
contradiction
 <b>by </b><i><a class="txt" href="kurato_1.html#E4:58"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="borsuk_5.html#D6" title="BORSUK_5:def.6">BORSUK_5:def 6</a>, <a class="ref" href="setfam_1.html#D9" title="SETFAM_1:def.9">SETFAM_1:def 9</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>


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