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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="setfam_1.html#V1">with_non-empty_elements</a> <a href="borsuk_5.html#V1">with_proper_subsets</a> <a href="struct_0.html#NM3">Subset-Family</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:58"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="tops_2.html#V1">open</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="tops_2.html#V2">closed</a> )
 ;<br/>

<b>assume </b><a NAME="E2:58"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:58"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E5:58"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="kurato_1.html#E4:58"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E7:58"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="kurato_1.html#E1:58"><i><font color="Green">E45</font></i></a>, <a class="txt" href="kurato_1.html#E4:58"><i><font color="Green">E46</font></i></a>, <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<a NAME="E8:58"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="kurato_1.html#E1:58"><i><font color="Green">E45</font></i></a>, <a class="txt" href="kurato_1.html#E5:58"><i><font color="Green">E47</font></i></a>, <a class="ref" href="tops_2.html#D2">TOPS_2:def 2</a></i>;<br/>
<a NAME="E9:58"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i><a class="txt" href="kurato_1.html#E7:58"><i><font color="Green">E48</font></i></a>, <a class="ref" href="borsuk_5.html#T57">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">E46</font></i></a>, <a class="ref" href="borsuk_5.html#D6">BORSUK_5:def 6</a>, <a class="ref" href="setfam_1.html#D9">SETFAM_1:def 9</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>


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