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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> implies  <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> )</span><br/>

<b>assume </b><a NAME="E1:42"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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></span><br/>

<a NAME="E2:42"/><b>then </b>
<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>then 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>  ;<br/>
<a NAME="E4:42"/><i><font color="Green" title="E41">A1</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="E5:42"/><i><font color="Green" title="E42">A2</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#K7" title="RFUNCT_1:func.7">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="E6:42"/>
(  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</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="E7:42"/><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="E8:42"/>
 <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="#E4:42"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="#E5:42"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="integra4.html#T2" title="INTEGRA4:th.2">INTEGRA4:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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