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<b>let </b><font color="Maroon" title="c1">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>







<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c2">A</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"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">a1</font>, <font color="Maroon" title="c6">b1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:26"/><i><font color="Green" title="E22">A2</font></i>: 
( <font color="Maroon" title="c5">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b1</font> &amp; <font color="Maroon" title="c2">A</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"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 <b>by </b><i><a class="ref" href="integra1.html#D1" title="INTEGRA1:def.1">INTEGRA1:def 1</a></i>;<br/>
<a NAME="E4:26"/><i><font color="Green" title="E23">A3</font></i>: 
( <font color="Maroon" title="c5">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> )
 
<b>by </b><i><a class="txt" href="integra5.html#E1:26"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="integra5.html#E3:26"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="integra1.html#T6" title="INTEGRA1:th.6">INTEGRA1:6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_1"/>
( <font color="Maroon" title="c4">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">a</font> or <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> )
 </span><b>by </b><i><a class="txt" href="integra5.html#E3:26"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="integra5.html#E4:26"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1"><b>suppose </b></a><a NAME="E1:26_1_1_1"/><i><font color="Green" title="E24">A4</font></i>: 
<font color="Maroon" title="c4">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">a</font>
 ;<br/><div class="add"><a NAME="E2:26_1_1_1"/><b>then </b>
 <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <font color="Maroon" title="c1">f</font>,<span class="p2"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="integra5.html#D5" target="_self" title="INTEGRA5:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E3:26_1_1_1"/>
 <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="txt" href="integra5.html#E1:26"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="integra5.html#E1:26_1_1_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="integra5.html#D4" target="_self" title="INTEGRA5:def.4">Def4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_2"><b>suppose </b></a><a NAME="E1:26_1_1_2"/><i><font color="Green" title="E24">A5</font></i>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 ;<br/><div class="add"><a NAME="E2:26_1_1_2"/><b>then </b><i><font color="Green" title="E25">A6</font></i>: 
 <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <font color="Maroon" title="c1">f</font>,<span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span>
 <b>by </b><i><a class="ref" href="integra5.html#D5" target="_self" title="INTEGRA5:def.5">Def5</a></i>;<br/><a NAME="E3:26_1_1_2"/>
<font color="Maroon" title="c2">A</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"><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" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/><a NAME="E4:26_1_1_2"/><b>then </b>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> )
 <b>by </b><i><a class="txt" href="integra5.html#E1:26"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="integra1.html#T6" title="INTEGRA1:th.6">INTEGRA1:6</a></i>;<br/><b>then </b> <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c2">A</font> = 
<span class="p1">(<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> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<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>
<b>by </b><i><a class="txt" href="integra5.html#E1:26_1_1_2"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><a NAME="E6:26_1_1_2"/><b>then </b>
 <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="integra4.html#T6" title="INTEGRA4:th.6">INTEGRA4:6</a></i>;<br/><b>hence </b><a NAME="E7:26_1_1_2"/>
 <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="txt" href="integra5.html#E1:26"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="integra5.html#E1:26_1_1_2"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="integra5.html#E2:26_1_1_2"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="integra5.html#D4" target="_self" title="INTEGRA5:def.4">Def4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:26"/>
 <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>
 ;<br/>


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