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<span class="kw">let </span><font color="Maroon" title="c1">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  implies  <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <font color="Maroon" title="c1">P</font> is  <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:18"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<font color="Maroon" title="c1">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <font color="Maroon" title="c1">P</font> is  <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:18_1_1"/>
(  <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1"><span class="kw">case </span></a><a NAME="E1:18_1_1_1"/>
 <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <font color="Maroon" title="c1">P</font> is  <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a> </span><br/><div class="add"><a NAME="E2:18_1_1_1"/>
 <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <font color="Maroon" title="c1">P</font> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="weierstr.html#T17" target="_self" title="WEIERSTR:th.17">Th17</a></span>;<br/><span class="kw">hence </span><a NAME="E3:18_1_1_1"/>
 <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <font color="Maroon" title="c1">P</font> is  <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="weierstr.html#T18" target="_self" title="WEIERSTR:th.18">Th18</a>, <a class="ref" href="rcomp_1.html#T29" title="RCOMP_1:th.29">RCOMP_1:29</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2"><span class="kw">case </span></a><a NAME="E1:18_1_1_2"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
 <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not  <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <font color="Maroon" title="c1">P</font> is  <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a>  implies  <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <font color="Maroon" title="c1">P</font> is  <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a>  )</span><br/><div class="add"><span class="kw">assume </span><a NAME="E2:18_1_1_2"/>
not  <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <font color="Maroon" title="c1">P</font> is  <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <font color="Maroon" title="c1">P</font> is  <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a> </span><br/><a NAME="E3:18_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A3</font></span>: 
 ex <font color="Olive" title="b1">s1</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> st <br/>(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <font color="Maroon" title="c1">P</font> &amp; ( for <font color="Olive" title="b2">s2</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  holds <br/>(  not <font color="Olive" title="b2">s2</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Olive" title="b1">s1</font> or  not <font color="Olive" title="b2">s2</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  or  not  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b2">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <font color="Maroon" title="c1">P</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D3" title="RCOMP_1:def.3">RCOMP_1:def 3</a></span>;<br/><a NAME="E4:18_1_1_2"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:18_1_1_2"/>
 <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <font color="Maroon" title="c1">P</font> is  <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1_1_2"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E3:18_1_1_2"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:18"/>
 <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <font color="Maroon" title="c1">P</font> is  <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a> 
 ; <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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