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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> <br/> for <font color="Olive" title="b2">x</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X9</font> &amp; <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/>(  <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Olive" title="b1">X9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">x</font> &amp; <font color="Olive" title="b2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Olive" title="b1">X9</font> )</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"> for <font color="Olive" title="b1">x</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X9</font> &amp; <font color="Maroon" title="c2">X9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> holds <br/>(  <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c2">X9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c2">X9</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X9</font> &amp; <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 (  <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c2">X9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c2">X9</font> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:133"/><span class="lab"><font color="Green" title="E59">A1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X9</font>
 <span class="kw">and </span><br/><a NAME="E2:133"/><span class="lab"><font color="Green" title="E60">A2</font></span>: 
<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"> (  <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c2">X9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c2">X9</font> )</span><br/>

<a NAME="E3:133"/>
( <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="txt" href="#E2:133"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="ref" href="borsuk_5.html#T107" target="_self" title="BORSUK_5:th.107">Th107</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:133"/>
(  <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c2">X9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c2">X9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</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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