<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <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>   st <font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a>  &amp; <font color="Olive" title="b2">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a>  holds <br/>( <font color="Olive" title="b1">f</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Olive" title="b2">g</font> is  <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a>  &amp;  <a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <font color="Olive" title="b1">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#K12" title="INTEGRA1:func.12">integral</a> <font color="Olive" title="b2">g</font></span>)</span> )</span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Maroon" title="c2">f</font> is  <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a>  &amp; <font color="Maroon" title="c3">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Maroon" title="c3">g</font> is  <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a>  implies ( <font color="Maroon" title="c2">f</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">g</font> is  <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a>  &amp;  <a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</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#K12" title="INTEGRA1:func.12">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#K12" title="INTEGRA1:func.12">integral</a> <font color="Maroon" title="c3">g</font></span>)</span> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:36"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
( <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Maroon" title="c2">f</font> is  <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a>  )
 <span class="kw">and </span><br/><a NAME="E2:36"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<font color="Maroon" title="c3">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 <span class="kw">and </span><br/><a NAME="E3:36"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
<font color="Maroon" title="c3">g</font> is  <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">g</font> is  <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a>  &amp;  <a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</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#K12" title="INTEGRA1:func.12">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#K12" title="INTEGRA1:func.12">integral</a> <font color="Maroon" title="c3">g</font></span>)</span> )</span><br/>

<a NAME="E4:36"/>
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#R2" title="FUNCT_2: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="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c3">g</font>
 
;<br/>
<a NAME="E5:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A4</font></span>: 
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">g</font> is  <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="integra2.html#T31" target="_self" title="INTEGRA2:th.31">Th31</a></span>;<br/>
<a NAME="E6:36"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
<span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="rfunct_1.html#T99" title="RFUNCT_1:th.99">RFUNCT_1:99</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:36"/>
<font color="Maroon" title="c2">f</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">g</font> is  <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E5:36"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="integra1.html#T59" title="INTEGRA1:th.59">INTEGRA1:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</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#K12" title="INTEGRA1:func.12">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#K12" title="INTEGRA1:func.12">integral</a> <font color="Maroon" title="c3">g</font></span>)</span></span><br/>

<a NAME="E8:36"/>
 <a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</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#K12" title="INTEGRA1:func.12">integral</a> <font color="Maroon" title="c3">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="integra2.html#T31" target="_self" title="INTEGRA2:th.31">Th31</a></span>;<br/>
<a NAME="E9:36"/><span class="kw">then </span>
 <a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</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#K12" title="INTEGRA1:func.12">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#K12" title="INTEGRA1:func.12">integral</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E6:36"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E5:36"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="integra1.html#T59" title="INTEGRA1:th.59">INTEGRA1:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:36"/>
 <a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</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#K12" title="INTEGRA1:func.12">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#K12" title="INTEGRA1:func.12">integral</a> <font color="Maroon" title="c3">g</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
