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<b>let </b><font color="Maroon" title="c1">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="c2">f</font>, <font color="Maroon" title="c3">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:36"/><i><font color="Green" title="E26">A1</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">f</font> <a href="integra1.html#R3" title="INTEGRA1:pred.3">is_integrable_on</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">g</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">g</font> <a href="integra1.html#R3" title="INTEGRA1:pred.3">is_integrable_on</a> <font color="Maroon" title="c1">A</font> )
 ;<br/>

<a NAME="E2:36"/>
 <a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c3">g</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T68" title="RFUNCT_1:th.68">RFUNCT_1:68</a></i>;<br/>
<a NAME="E3:36"/><b>then </b><i><font color="Green" title="E27">A2</font></i>: 
 <a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c3">g</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T131" title="FUNCT_2:th.131">FUNCT_2:131</a></i>;<br/>
<a NAME="E4:36"/><i><font color="Green" title="E28">A3</font></i>: 
 <a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c3">g</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="integra2.html#E1:36"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="rfunct_1.html#T99" title="RFUNCT_1:th.99">RFUNCT_1:99</a></i>;<br/>
<a NAME="E5:36"/>
 <a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">g</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T38" title="RFUNCT_1:th.38">RFUNCT_1:38</a></i>;<br/>
<a NAME="E6:36"/><b>then </b><i><font color="Green" title="E29">A4</font></i>: 
(  <a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c3">g</font> <a href="integra1.html#R3" title="INTEGRA1:pred.3">is_integrable_on</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c3">g</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra2.html#E1:36"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="integra2.html#T31" target="_self" title="INTEGRA2:th.31">Th31</a></i>;<br/>
<a NAME="E7:36"/><i><font color="Green" title="E30">A5</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16" title="SEQ_1:func.16">-</a> <font color="Maroon" title="c3">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T40" title="RFUNCT_1:th.40">RFUNCT_1:40</a></i>;<br/>
<b>hence </b><a NAME="E8:36"/>
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c3">g</font> <a href="integra1.html#R3" title="INTEGRA1:pred.3">is_integrable_on</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="integra2.html#E1:36"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="integra2.html#E3:36"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="integra2.html#E4:36"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="integra2.html#E6:36"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="integra1.html#T59" title="INTEGRA1:th.59">INTEGRA1:59</a></i>;<br/>

<a NAME="E9:36"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <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="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra2.html#E1:36"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="integra2.html#E3:36"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="integra2.html#E4:36"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="integra2.html#E6:36"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="integra2.html#E7:36"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="integra1.html#T59" title="INTEGRA1:th.59">INTEGRA1:59</a></i>;<br/>
<b>hence </b><a NAME="E10:36"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c3">g</font></span>)</span>
 ;<br/>


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