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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">P</font>, <font color="Olive" title="b2">Q</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">P</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> &amp; <font color="Olive" title="b2">Q</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> &amp;  <a href="hausdorf.html#K3" title="HAUSDORF:func.3">HausDist</a> <font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Q</font></span><br/><b>let </b><font color="Maroon" title="c2">P</font>, <font color="Maroon" title="c3">Q</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">P</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> &amp; <font color="Maroon" title="c3">Q</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> &amp;  <a href="hausdorf.html#K3" title="HAUSDORF:func.3">HausDist</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Q</font> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:47"/><i><font color="Green" title="E35">A1</font></i>: 
( <font color="Maroon" title="c2">P</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> &amp; <font color="Maroon" title="c3">Q</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> )
 <b>and </b><br/><a NAME="E2:47"/><i><font color="Green" title="E36">A2</font></i>: 
 <a href="hausdorf.html#K3" title="HAUSDORF:func.3">HausDist</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Q</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c4">P1</font> = <font color="Maroon" title="c2">P</font>, <font color="Maroon" title="c5">Q1</font> = <font color="Maroon" title="c3">Q</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> ;<br/>
<a NAME="E4:47"/>
 <a href="hausdorf.html#K1" title="HAUSDORF:func.1">HausDist</a> <font color="Maroon" title="c4">P1</font>,<font color="Maroon" title="c5">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E2:47"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="hausdorf.html#D3" target="_self" title="HAUSDORF:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E5:47"/>
<font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Q</font>
 <b>by </b><i><a class="txt" href="#E1:47"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="hausdorf.html#T39" target="_self" title="HAUSDORF:th.39">Th39</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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