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<b>let </b><font color="Maroon" title="c1">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be  <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> ;<br/><b>let </b><font color="Maroon" title="c3">S</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="integra1.html#M2" title="INTEGRA1:mode.2">Division</a> of <font color="Maroon" title="c2">A</font>;<br/><b>let </b><font color="Maroon" title="c4">D</font> be    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of <font color="Maroon" title="c3">S</font>;<br/>







<b>assume </b><a NAME="E1:60"/><i><font color="Green" title="E41">A1</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D</font>
 ;<br/>

<a NAME="E2:60"/><i><font color="Green" title="E42">A2</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D2" target="_self" title="INTEGRA1:def.2">Def2</a></i>;<br/>
<a NAME="E3:60"/>
<font color="Maroon" title="c4">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:60"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:60"/><b>then </b>
<font color="Maroon" title="c4">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:60"><i><font color="Green" title="E42">A2</font></i></a></i>;<br/>
<a NAME="E5:60"/><b>then </b>
<font color="Maroon" title="c4">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c2">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th5</a></i>;<br/>
<a NAME="E6:60"/><b>then </b>
<font color="Maroon" title="c4">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c2">A</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E7:60"/><b>then </b><i><font color="Green" title="E43">A3</font></i>: 
 ex <font color="Olive" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c1">i</font> &amp;  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c2">A</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">A1</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="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c1">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> 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>  <b>by </b><i><a class="ref" href="integra1.html#D1" target="_self" title="INTEGRA1:def.1">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">A1</font>
;<br/>

<b>reconsider </b><font color="Maroon" title="c6">A2</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"><font color="Maroon" title="c4">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c2">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> 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>  <b>by </b><i><a class="txt" href="integra1.html#E7:60"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="integra1.html#D1" target="_self" title="INTEGRA1:def.1">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">A2</font>
;<br/>

<font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">A2</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="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c2">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E7:60"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="rcomp_1.html#T50" title="RCOMP_1:th.50">RCOMP_1:50</a></i>
<br/>.= 
<font color="Maroon" title="c2">A</font>
<b>by </b><i><a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th5</a></i>
;<br/>
<b>hence </b><a NAME="E11:60"/>
( <font color="Maroon" title="c5">A1</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="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c1">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c6">A2</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"><font color="Maroon" title="c4">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c2">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">A2</font> )
 ;<br/>


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