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

<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <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#K2" title="NUMBERS:func.2">COMPLEX</a><br/>  for <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#K2" title="NUMBERS:func.2">COMPLEX</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> holds <br/>( <font color="Olive" title="b1">f</font> <a href="integr16.html#R1" title="INTEGR16:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font> iff <font color="Olive" title="b2">g</font> is  <a href="integr16.html#V1" title="INTEGR16:attr.1">integrable</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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#K2" title="NUMBERS:func.2">COMPLEX</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> holds <br/>( <font color="Maroon" title="c2">f</font> <a href="integr16.html#R1" title="INTEGR16:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font> iff <font color="Olive" title="b1">g</font> is  <a href="integr16.html#V1" title="INTEGR16:attr.1">integrable</a>  )</span><br/><span class="kw">let </span><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#K2" title="NUMBERS:func.2">COMPLEX</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> implies ( <font color="Maroon" title="c2">f</font> <a href="integr16.html#R1" title="INTEGR16:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font> iff <font color="Maroon" title="c3">g</font> is  <a href="integr16.html#V1" title="INTEGR16:attr.1">integrable</a>  ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:28"/><span class="lab"><font color="Green" title="E17">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="integr16.html#R1" title="INTEGR16:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font> iff <font color="Maroon" title="c3">g</font> is  <a href="integr16.html#V1" title="INTEGR16:attr.1">integrable</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E2:28"/>
( <font color="Maroon" title="c2">f</font> <a href="integr16.html#R1" title="INTEGR16:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font> implies <font color="Maroon" title="c3">g</font> is  <a href="integr16.html#V1" title="INTEGR16:attr.1">integrable</a>  )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">g</font> is  <a href="integr16.html#V1" title="INTEGR16:attr.1">integrable</a>  implies <font color="Maroon" title="c2">f</font> <a href="integr16.html#R1" title="INTEGR16:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="28_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:28_1"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="integr16.html#R1" title="INTEGR16:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">g</font> is  <a href="integr16.html#V1" title="INTEGR16:attr.1">integrable</a> </span><br/>

<a NAME="E2:28_1"/>
(  <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c3">g</font> is  <a href="integra1.html#V3" title="INTEGRA1:attr.3">integrable</a>  &amp;  <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c3">g</font> is  <a href="integra1.html#V3" title="INTEGRA1:attr.3">integrable</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="28_1_1"><span class="kw">proof </span></a><div class="add">

<span class="lab"><font color="Green" title="E19">A3</font></span>: <font color="Maroon" title="c1">A</font> = 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
;<br/>
<a NAME="E2:28_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D3" onmouseover="rs('comseq_3/D3')" onmouseout="rh()">COMSEQ_3:def 3</a></span>;<br/>
<a NAME="E3:28_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra5.html#T6" onmouseover="rs('integra5/T6')" onmouseout="rh()">INTEGRA5:6</a>, <a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">F</font> = <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> as   <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/>
<a NAME="E5:28_1_1"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
 <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a></span>;<br/>
<a NAME="E6:28_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">g0</font> = <font color="Maroon" title="c3">g</font> as   <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#K2" title="NUMBERS:func.2">COMPLEX</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T5" onmouseover="rs('relset_1/T5')" onmouseout="rh()">RELSET_1:5</a></span>;<br/>
 <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c3">g</font> = 
 <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c5">g0</font>

<br/>.= 
<font color="Maroon" title="c4">F</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="integr16.html#E27"><span class="lab"><font color="Green" title="E16">Lm4</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:28_1_1"/>
 <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c3">g</font> is  <a href="integra1.html#V3" title="INTEGRA1:attr.3">integrable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28_1_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c3">g</font> is  <a href="integra1.html#V3" title="INTEGRA1:attr.3">integrable</a> </span><br/>

<a NAME="E10:28_1_1"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="comseq_3.html#D4" onmouseover="rs('comseq_3/D4')" onmouseout="rh()">COMSEQ_3:def 4</a></span>;<br/>
<a NAME="E11:28_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra5.html#T6" onmouseover="rs('integra5/T6')" onmouseout="rh()">INTEGRA5:6</a>, <a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">G</font> = <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> as   <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/>
<a NAME="E13:28_1_1"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
 <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a></span>;<br/>
 <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c3">g</font> = 
 <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c5">g0</font>

<br/>.= 
<font color="Maroon" title="c6">G</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="integr16.html#E27"><span class="lab"><font color="Green" title="E16">Lm4</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E15:28_1_1"/>
 <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c3">g</font> is  <a href="integra1.html#V3" title="INTEGRA1:attr.3">integrable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:28_1_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:28_1"/>
<font color="Maroon" title="c3">g</font> is  <a href="integr16.html#V1" title="INTEGR16:attr.1">integrable</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E3:28"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
<font color="Maroon" title="c3">g</font> is  <a href="integr16.html#V1" title="INTEGR16:attr.1">integrable</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="integr16.html#R1" title="INTEGR16:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font></span><br/>

<a NAME="E4:28"/>
(  <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="integr16.html#E27"><span class="lab"><font color="Green" title="E16">Lm4</font></span></a>, <a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E18">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:28"/>
<font color="Maroon" title="c2">f</font> <a href="integr16.html#R1" title="INTEGR16:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
