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<b>reconsider </b><font color="Maroon" title="c1">D</font> =  <a href="metric_1.html#K6" title="METRIC_1:func.6">DiscreteSpace</a> 2 as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c2">a</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <font color="Maroon" title="c3">b</font> = 1 as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">D</font> <b>by </b><i><a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a></i>;<br/>
<a NAME="E3:73"/><i><font color="Green" title="E55">A1</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D11" title="METRIC_1:def.11">METRIC_1:def 11</a></i>;<br/>
<a NAME="E4:73"/>
 <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">D</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 
<b>by </b><i><a class="ref" href="compts_1.html#T27" title="COMPTS_1:th.27">COMPTS_1:27</a></i>;<br/>
<a NAME="E5:73"/><b>then </b>
( <font color="Maroon" title="c1">D</font> is <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a> &amp; not  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
<b>by </b><i><a class="ref" href="tbsp_1.html#T10" title="TBSP_1:th.10">TBSP_1:10</a></i>;<br/>
<a NAME="E6:73"/><b>then </b>
(  <a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  is <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a> &amp;  ex <font color="Olive" title="b1">S</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="compl_sp.html#V1" title="COMPL_SP:attr.1">bounded</a> <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <span class="p1">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> st <br/>( <font color="Olive" title="b1">S</font> is <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a> &amp; <font color="Olive" title="b1">S</font> is <a href="kurato_2.html#V1" title="KURATO_2:attr.1">descending</a> &amp;  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Olive" title="b1">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )
 
<b>by </b><i><a class="txt" href="compl_sp.html#E3:73"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="compl_sp.html#T43" target="_self" title="COMPL_SP:th.43">Th43</a></i>;<br/>
<b>hence </b><a NAME="E7:73"/>
 ex <font color="Olive" title="b1">M</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a> st <br/>( <font color="Olive" title="b1">M</font> is <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a> &amp;  ex <font color="Olive" title="b2">S</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="compl_sp.html#V1" title="COMPL_SP:attr.1">bounded</a> <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <font color="Olive" title="b1">M</font> st <br/>( <font color="Olive" title="b2">S</font> is <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a> &amp; <font color="Olive" title="b2">S</font> is <a href="kurato_2.html#V1" title="KURATO_2:attr.1">descending</a> &amp;  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Olive" title="b2">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )
 ;<br/>


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