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<b>let </b><font color="Maroon">A</font> be  <a href="integra1.html#V1">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a> , <font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:36"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">f</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">g</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">g</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font> )
 ;<br/>

<a NAME="E2:36"/>
 <a href="seq_1.html#K16">-</a> <font color="Maroon">g</font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T68">RFUNCT_1:68</a></i>;<br/>
<a NAME="E3:36"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="seq_1.html#K16">-</a> <font color="Maroon">g</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T131">FUNCT_2:131</a></i>;<br/>
<a NAME="E4:36"/><i><font color="Green">E39</font></i>: 
 <a href="seq_1.html#K16">-</a> <font color="Maroon">g</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="rfunct_1.html#T99">RFUNCT_1:99</a></i>;<br/>
<a NAME="E5:36"/>
 <a href="seq_1.html#K16">-</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T38">RFUNCT_1:38</a></i>;<br/>
<a NAME="E6:36"/><b>then </b><i><font color="Green">E40</font></i>: 
(  <a href="seq_1.html#K16">-</a> <font color="Maroon">g</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font> &amp;  <a href="integra1.html#K13">integral</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <font color="Maroon">g</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:36"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T40">RFUNCT_1:40</a></i>;<br/>
<b>hence </b><a NAME="E8:36"/>
<font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">g</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font>
 <b>by </b><i>, , , , <a class="ref" href="integra1.html#T59">INTEGRA1:59</a></i>;<br/>

<a NAME="E9:36"/>
 <a href="integra1.html#K13">integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <font color="Maroon">f</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K13">integral</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 
<b>by </b><i>, , , , , <a class="ref" href="integra1.html#T59">INTEGRA1:59</a></i>;<br/>
<b>hence </b><a NAME="E10:36"/>
 <a href="integra1.html#K13">integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <font color="Maroon">f</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <font color="Maroon">g</font></span>)</span>
 ;<br/>


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