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<span class="kw">let </span><font color="Maroon" title="c1">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.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">A</font> being  <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/> for <font color="Olive" title="b2">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>   st <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b2">f</font> <a href="integr16.html#R1" title="INTEGR16:pred.1">is_integrable_on</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  holds <br/>( <font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b2">f</font> <a href="integr16.html#R1" title="INTEGR16:pred.1">is_integrable_on</a> <font color="Olive" title="b1">A</font> &amp;  <a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b2">f</font></span>)</span>,<font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b1">A</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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> 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>   st <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> <a href="integr16.html#R1" title="INTEGR16:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  holds <br/>( <font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b1">f</font> <a href="integr16.html#R1" title="INTEGR16:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b1">f</font></span>)</span>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c2">A</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c3">f</font> <a href="integr16.html#R1" title="INTEGR16:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  implies ( <font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font> <a href="integr16.html#R1" title="INTEGR16:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">A</font></span>)</span> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:34"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E2:34"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="integr16.html#R1" title="INTEGR16:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E3:34"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</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="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font> <a href="integr16.html#R1" title="INTEGR16:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">A</font></span>)</span> )</span><br/>

<a NAME="E4:34"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
(  <a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <font color="Maroon" title="c3">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <font color="Maroon" title="c3">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="integr16.html#D9" target="_self" title="INTEGR16:def.9">Def16</a></span>;<br/>
<a NAME="E5:34"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
(  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p2">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">A</font></span>)</span> &amp;  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p2">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">A</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:34_1"/>
(  <a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font></span>)</span> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp;  <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font></span>)</span> 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="#E3:34"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="integr16.html#T11" target="_self" title="INTEGR16:th.11">BND01A</a></span>;<br/>
<a NAME="E2:34_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A51</font></span>: 
( <span class="p1">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <span class="p1">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">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="integr16.html#E29"><span class="lab"><font color="Green" title="E19">Lm6</font></span></a></span>;<br/>
<a NAME="E3:34_1"/><span class="lab"><font color="Green" title="E27">A52</font></span>: 
( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <font color="Maroon" title="c3">f</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="mesfun6c.html#D1" title="MESFUN6C:def.1">MESFUN6C:def 1</a>, <a class="ref" href="mesfun6c.html#D2" title="MESFUN6C:def.2">MESFUN6C:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:34_1"/>
 <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p2">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E2:34_1"><span class="lab"><font color="Green" title="E26">A51</font></span></a>, <a class="ref" href="integra6.html#T9" title="INTEGRA6:th.9">INTEGRA6:9</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="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p2">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">A</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E5:34_1"/>
 <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p2">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E2:34_1"><span class="lab"><font color="Green" title="E26">A51</font></span></a>, <a class="txt" href="#E3:34_1"><span class="lab"><font color="Green" title="E27">A52</font></span></a>, <a class="ref" href="integra6.html#T9" title="INTEGRA6:th.9">INTEGRA6:9</a></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><span class="kw">end;</span></div>
<a NAME="E6:34"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
(  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> &amp;  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p2">(<span class="default"><a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:34_2"/>
(  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p1">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p2">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">A</font></span>)</span> &amp;  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p1">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p2">(<span class="default"><a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> <span class="p2">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">A</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T28" title="COMPLEX1:th.28">COMPLEX1:28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:34_2"/>
(  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> &amp;  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p2">(<span class="default"><a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="mesfun6c.html#T2" title="MESFUN6C:th.2">MESFUN6C:2</a></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><span class="kw">end;</span></div>
<a NAME="E7:34"/>
(  <a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:34_3"/>
( <span class="p1">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="integr16.html#E29"><span class="lab"><font color="Green" title="E19">Lm6</font></span></a></span>;<br/>
<a NAME="E2:34_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A71</font></span>: 
( <span class="p1">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <span class="p1">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">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="#E3:34"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="integr16.html#T11" target="_self" title="INTEGR16:th.11">BND01A</a></span>;<br/>
<a NAME="E3:34_3"/><span class="lab"><font color="Green" title="E28">A72</font></span>: 
( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <font color="Maroon" title="c3">f</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="mesfun6c.html#D1" title="MESFUN6C:def.1">MESFUN6C:def 1</a>, <a class="ref" href="mesfun6c.html#D2" title="MESFUN6C:def.2">MESFUN6C:def 2</a></span>;<br/>
<a NAME="E4:34_3"/>
(  <a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <font color="Maroon" title="c3">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <font color="Maroon" title="c3">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="integr16.html#D9" target="_self" title="INTEGR16:def.9">Def16</a></span>;<br/>
<a NAME="E5:34_3"/><span class="kw">then </span>
( <font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_3"><span class="lab"><font color="Green" title="E27">A71</font></span></a>, <a class="txt" href="#E3:34_3"><span class="lab"><font color="Green" title="E28">A72</font></span></a>, <a class="ref" href="integra6.html#T9" title="INTEGRA6:th.9">INTEGRA6:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:34_3"/>
(  <a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun6c.html#T2" title="MESFUN6C:th.2">MESFUN6C:2</a></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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:34"/>
<font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font> <a href="integr16.html#R1" title="INTEGR16:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integr16.html#D9" target="_self" title="INTEGR16:def.9">Def16</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="integr16.html#K9" title="INTEGR16:func.9">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">A</font></span>)</span></span><br/>

<a NAME="E9:34"/>
(  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> &amp;  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="integr16.html#T1" target="_self" title="INTEGR16:th.1">LM001</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:34"/>
 <a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="integr16.html#K9" title="INTEGR16:func.9">integral</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#D5" title="COMPLEX1:def.5">COMPLEX1:def 5</a></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>
