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<b>let </b><font color="Maroon" title="c1">M</font> be  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>; <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> 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="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span><br/> for <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><br/> for <font color="Olive" title="b3">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K4" title="WEIERSTR:func.4">dist</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> holds <br/><font color="Olive" title="b3">X</font> is <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a></span><br/><b>let </b><font color="Maroon" title="c2">P</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="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>; <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">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><br/> for <font color="Olive" title="b2">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K4" title="WEIERSTR:func.4">dist</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c2">P</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> holds <br/><font color="Olive" title="b2">X</font> is <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a></span><br/><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>; <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">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K4" title="WEIERSTR:func.4">dist</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c2">P</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> holds <br/><font color="Olive" title="b1">X</font> is <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a></span><br/><b>let </b><font color="Maroon" title="c4">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K4" title="WEIERSTR:func.4">dist</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c2">P</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> implies <font color="Maroon" title="c4">X</font> is <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> )</span><br/>







<b>assume </b><a NAME="E1:23"/><i><font color="Green" title="E20">A1</font></i>: 
( <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K4" title="WEIERSTR:func.4">dist</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c2">P</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">X</font> is <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a></span><br/>

<a NAME="E2:23"/><b>then </b><i><font color="Green" title="E21">A2</font></i>: 
 <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K4" title="WEIERSTR:func.4">dist</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font></span>)</span> is <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a>
 
<b>by </b><i><a class="ref" href="weierstr.html#T15" title="WEIERSTR:th.15">WEIERSTR:15</a>, <a class="ref" href="weierstr.html#T17" title="WEIERSTR:th.17">WEIERSTR:17</a></i>;<br/>
<a NAME="E3:23"/>
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K4" title="WEIERSTR:func.4">dist</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:23"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="weierstr.html#D3" title="WEIERSTR:def.3">WEIERSTR:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:23"/>
<font color="Maroon" title="c4">X</font> is <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>
 <b>by </b><i><a class="txt" href="#E2:23"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="xxreal_2.html#D11" title="XXREAL_2:def.11">XXREAL_2:def 11</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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