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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:45"/>
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 ;<br/>

<a NAME="E2:45"/><b>then </b><i><font color="Green" title="E44">A1</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span>
 
<b>by </b><i><a class="ref" href="integra5.html#D4" title="INTEGRA5:def.4">INTEGRA5:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">A</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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="integra8.html#E2:45"><i><font color="Green" title="E44">A1</font></i></a></i>;<br/>
<a NAME="E4:45"/><i><font color="Green" title="E45">A2</font></i>: 
 <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <span class="p1">(<span class="default"><a href="integra8.html#K1" title="INTEGRA8:func.1">Cst</a> 1</span>)</span>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p1">(<span class="default"><a href="integra8.html#K1" title="INTEGRA8:func.1">Cst</a> 1</span>)</span>,<font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="integra5.html#T19" title="INTEGRA5:th.19">INTEGRA5:19</a></i>;<br/>
<a NAME="E5:45"/><i><font color="Green" title="E46">A3</font></i>: 
<span class="p1">(<span class="default"><a href="integra8.html#K1" title="INTEGRA8:func.1">Cst</a> 1</span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="integra8.html#T36" target="_self" title="INTEGRA8:th.36">Th36</a></i>;<br/>
<a NAME="E6:45"/>
(  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp;  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> )
 
<b>by </b><i><a class="ref" href="integra8.html#T37" target="_self" title="INTEGRA8:th.37">Th37</a></i>;<br/>
<a NAME="E7:45"/><b>then </b>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E8:45"/>
 <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <span class="p1">(<span class="default"><a href="integra8.html#K1" title="INTEGRA8:func.1">Cst</a> 1</span>)</span>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font>
 <b>by </b><i><a class="txt" href="integra8.html#E4:45"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="integra8.html#E5:45"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="integra4.html#T2" title="INTEGRA4:th.2">INTEGRA4:2</a></i>;<br/>


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