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<span class="kw">reconsider </span><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/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/>
<a NAME="E3:73"/>
 <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">D</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="compts_1.html#T27" title="COMPTS_1:th.27">COMPTS_1:27</a></span>;<br/>
<a NAME="E4:73"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A1</font></span>: 
<font color="Maroon" title="c1">D</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tbsp_1.html#T10" title="TBSP_1:th.10">TBSP_1:10</a></span>;<br/>
<a NAME="E5:73"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D11" title="METRIC_1:def.11">METRIC_1:def 11</a></span>;<br/>
<a NAME="E6:73"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A3</font></span>: 
 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="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</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>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="compl_sp.html#T43" target="_self" title="COMPL_SP:th.43">Th43</a></span>;<br/>
<a NAME="E7:73"/>
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="txt" href="#E4:73"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="compl_sp.html#T43" target="_self" title="COMPL_SP:th.43">Th43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:73"/>
 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="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</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>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73"><span class="lab"><font color="Green" title="E57">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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