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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  <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"> for <font color="Olive" title="b1">X9</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">X9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> holds <br/>( <font color="Olive" title="b1">X9</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>  &amp; <font color="Olive" title="b1">X9</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">X9</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">X9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> implies ( <font color="Maroon" title="c2">X9</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>  &amp; <font color="Maroon" title="c2">X9</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:132"/>
<font color="Maroon" title="c2">X9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 ; <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="c2">X9</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>  &amp; <font color="Maroon" title="c2">X9</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>  )</span><br/>

<a NAME="E2:132"/><span class="kw">then </span>
<font color="Maroon" title="c2">X9</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="ref" href="topreal6.html#T81" title="TOPREAL6:th.81">TOPREAL6:81</a></span>;<br/>
<a NAME="E3:132"/><span class="kw">then </span>
<font color="Maroon" title="c2">X9</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="ref" href="rcomp_1.html#T28" title="RCOMP_1:th.28">RCOMP_1:28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:132"/>
( <font color="Maroon" title="c2">X9</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>  &amp; <font color="Maroon" title="c2">X9</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D11" title="XXREAL_2:def.11">XXREAL_2:def 11</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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