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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">d</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</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>assume </b><a NAME="E1:33"/><i><font color="Green" title="E31">A1</font></i>: 
( <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> &amp; <font color="Maroon" title="c5">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</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> &amp; <font color="Maroon" title="c6">g</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</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> &amp; <font color="Maroon" title="c5">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</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> &amp; <font color="Maroon" title="c6">g</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</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> &amp; <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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> &amp; <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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> &amp; <font color="Maroon" title="c4">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1"/><i><font color="Green" title="E32">A2</font></i>: 
not <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font>
 ;<br/><a NAME="E2:33_1"/><b>then </b><i><font color="Green" title="E33">A3</font></i>: 
(  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <font color="Maroon" title="c5">f</font>,<span class="p2"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">c</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span></span>)</span> &amp;  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <font color="Maroon" title="c6">g</font>,<span class="p2"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">c</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span></span>)</span> &amp;  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c6">g</font></span>)</span>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c6">g</font></span>)</span>,<span class="p2"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">c</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span></span>)</span> &amp;  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c6">g</font></span>)</span>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c6">g</font></span>)</span>,<span class="p2"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">c</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" title="INTEGRA5:def.5">INTEGRA5:def 5</a></i>;<br/><b>hence </b> <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c6">g</font></span>)</span>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> = 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c6">g</font></span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">c</font></span>)</span>
<b>by </b><i><a class="txt" href="integra6.html#E1:33_1"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="integra5.html#D5" title="INTEGRA5:def.5">INTEGRA5:def 5</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra6.html#E1:33"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="integra6.html#E1:33_1"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="integra6.html#E34"><i><font color="Green" title="E29">Lm11</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">c</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">c</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">c</font></span>)</span>
<b>by </b><i><a class="txt" href="integra6.html#E1:33_1"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="integra6.html#E2:33_1"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="integra5.html#D5" title="INTEGRA5:def.5">INTEGRA5:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
<b>by </b><i><a class="txt" href="integra6.html#E1:33_1"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="integra6.html#E2:33_1"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="integra5.html#D5" title="INTEGRA5:def.5">INTEGRA5:def 5</a></i>
;<br/><br/><b>thus </b> <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c6">g</font></span>)</span>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> = 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c6">g</font></span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">c</font></span>)</span>
<b>by </b><i><a class="txt" href="integra6.html#E1:33_1"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="integra6.html#E2:33_1"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="integra5.html#D5" title="INTEGRA5:def.5">INTEGRA5:def 5</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra6.html#E1:33"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="integra6.html#E1:33_1"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="integra6.html#E34"><i><font color="Green" title="E29">Lm11</font></i></a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra6.html#E1:33_1"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="integra6.html#E2:33_1"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="integra5.html#D5" title="INTEGRA5:def.5">INTEGRA5:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">c</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
<b>by </b><i><a class="txt" href="integra6.html#E1:33_1"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="integra6.html#E2:33_1"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="integra5.html#D5" title="INTEGRA5:def.5">INTEGRA5:def 5</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:33"/>
(  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c6">g</font></span>)</span>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> &amp;  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c6">g</font></span>)</span>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra6.html#E1:33"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="integra6.html#E34"><i><font color="Green" title="E29">Lm11</font></i></a></i>;<br/>


</div>
