<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">C</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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#K5" title="TOPMETR:func.5">I[01]</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">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> implies  ex <font color="Olive" title="b1">D</font> being  <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a>  <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 <br/>( <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp;  <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Olive" title="b1">D</font> &amp;  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Olive" title="b1">D</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:86"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
<font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">D</font> being  <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a>  <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 <br/>( <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp;  <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Olive" title="b1">D</font> &amp;  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Olive" title="b1">D</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c2">C9</font> = <font color="Maroon" title="c1">C</font> as   <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">D</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c2">C9</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c2">C9</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <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/>
<a NAME="E4:86"/><span class="lab"><font color="Green" title="E53">A2</font></span>: 
( <font color="Maroon" title="c2">C9</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>  &amp; <font color="Maroon" title="c2">C9</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_4.html#T50" target="_self" title="BORSUK_4:th.50">Th50</a></span>;<br/>
<a NAME="E5:86"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A3</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">C9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">C9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T24" title="SEQ_4:th.24">SEQ_4:24</a></span>;<br/>
<a NAME="E6:86"/><span class="lab"><font color="Green" title="E55">A4</font></span>: 
<font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:86_1"/><span class="lab"><font color="Green" title="E56">A5</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font></span><br/>

<a NAME="E2:86_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">x9</font> = <font color="Maroon" title="c4">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></span>;<br/>
<a NAME="E4:86_1"/><span class="lab"><font color="Green" title="E57">A6</font></span>: 
<font color="Maroon" title="c5">x9</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">C9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="ref" href="borsuk_4.html#T51" target="_self" title="BORSUK_4:th.51">Th51</a></span>;<br/>
<a NAME="E5:86_1"/>
 <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c2">C9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="ref" href="borsuk_4.html#T51" target="_self" title="BORSUK_4:th.51">Th51</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:86_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:86_1"><span class="lab"><font color="Green" title="E57">A6</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</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>
<a NAME="E7:86"/><span class="lab"><font color="Green" title="E56">A7</font></span>: 
<font color="Maroon" title="c2">C9</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_4.html#T52" target="_self" title="BORSUK_4:th.52">Th52</a></span>;<br/>
<a NAME="E8:86"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A8</font></span>: 
 <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c2">C9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_4.html#T50" target="_self" title="BORSUK_4:th.50">Th50</a>, <a class="ref" href="rcomp_1.html#T31" title="RCOMP_1:th.31">RCOMP_1:31</a></span>;<br/>
<a NAME="E9:86"/><span class="lab"><font color="Green" title="E58">A9</font></span>: 
 <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c2">C9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:86"><span class="lab"><font color="Green" title="E56">A7</font></span></a>, <a class="ref" href="borsuk_4.html#T50" target="_self" title="BORSUK_4:th.50">Th50</a>, <a class="ref" href="rcomp_1.html#T30" title="RCOMP_1:th.30">RCOMP_1:30</a></span>;<br/>
<a NAME="E10:86"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A10</font></span>: 
<font color="Maroon" title="c3">D</font> is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>   <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#K5" title="TOPMETR:func.5">I[01]</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:86"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E8:86"><span class="lab"><font color="Green" title="E57">A8</font></span></a>, <a class="ref" href="borsuk_4.html#T49" target="_self" title="BORSUK_4:th.49">Th49</a>, <a class="ref" href="seq_4.html#T24" title="SEQ_4:th.24">SEQ_4:24</a></span>;<br/>
<a NAME="E11:86"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A11</font></span>: 
<font color="Maroon" title="c3">D</font> is   <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a>  <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_4.html#T55" target="_self" title="BORSUK_4:th.55">Th55</a></span>;<br/>
<a NAME="E12:86"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A12</font></span>: 
<font color="Maroon" title="c3">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c3">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c3">D</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></span>;<br/>
<a NAME="E13:86"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A13</font></span>: 
 <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c2">C9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c3">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:86"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="ref" href="xxreal_1.html#T66" title="XXREAL_1:th.66">XXREAL_1:66</a></span>;<br/>
<a NAME="E14:86"/><span class="lab"><font color="Green" title="E63">A14</font></span>: 
not 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:86_2"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:86_2"/><span class="kw">then </span>
 <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c3">D</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:86"><span class="lab"><font color="Green" title="E60">A11</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:86_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E9:86"><span class="lab"><font color="Green" title="E58">A9</font></span></a>, <a class="txt" href="#E13:86"><span class="lab"><font color="Green" title="E62">A13</font></span></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</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>
<a NAME="E15:86"/><span class="lab"><font color="Green" title="E64">A15</font></span>: 
 <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c2">C9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c3">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:86"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E12:86"><span class="lab"><font color="Green" title="E61">A12</font></span></a>, <a class="ref" href="xxreal_1.html#T66" title="XXREAL_1:th.66">XXREAL_1:66</a></span>;<br/>
<a NAME="E16:86"/><span class="lab"><font color="Green" title="E65">A16</font></span>: 
not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:86_3"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:86_3"/><span class="kw">then </span>
 <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c3">D</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:86"><span class="lab"><font color="Green" title="E60">A11</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:86_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E8:86"><span class="lab"><font color="Green" title="E57">A8</font></span></a>, <a class="txt" href="#E15:86"><span class="lab"><font color="Green" title="E64">A15</font></span></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</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>
<a NAME="E17:86"/><span class="lab"><font color="Green" title="E66">A17</font></span>: 
<font color="Maroon" title="c3">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:86_4"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span><br/>

<span class="kw">hence </span><a NAME="E2:86_4"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:86"><span class="lab"><font color="Green" title="E59">A10</font></span></a>, <a class="txt" href="#E16:86"><span class="lab"><font color="Green" title="E65">A16</font></span></a>, <a class="txt" href="#E14:86"><span class="lab"><font color="Green" title="E63">A14</font></span></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="xxreal_1.html#T5" title="XXREAL_1:th.5">XXREAL_1:5</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/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c4">D</font> = <font color="Maroon" title="c3">D</font> as   <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a>  <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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:86"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="ref" href="integra1.html#D1" title="INTEGRA1:def.1">INTEGRA1:def 1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">D</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="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c4">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp;  <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c4">D</font> &amp;  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c4">D</font> )</span><br/>

<span class="kw">thus </span><a NAME="E19:86"/>
( <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c4">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp;  <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c4">D</font> &amp;  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c4">D</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:86"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="txt" href="#E5:86"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E12:86"><span class="lab"><font color="Green" title="E61">A12</font></span></a>, <a class="txt" href="#E17:86"><span class="lab"><font color="Green" title="E66">A17</font></span></a>, <a class="ref" href="xxreal_1.html#T66" title="XXREAL_1:th.66">XXREAL_1:66</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/>

<span class="kw">consider </span><font color="Maroon" title="c5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C9</font>;<br/>

</div>
