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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/>  for <font color="Olive" title="b2">A</font> being   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><br/>  for <font color="Olive" title="b3">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>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b2">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="b3">f</font> &amp; <font color="Olive" title="b3">f</font> <a href="integr15.html#R1" title="INTEGR15:pred.1">is_integrable_on</a> <font color="Olive" title="b2">A</font> &amp; <font color="Olive" title="b3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b2">A</font> is  <a href="integr15.html#V3" title="INTEGR15:attr.3">bounded</a>  holds <br/>( <font color="Olive" title="b1">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Olive" title="b3">f</font> <a href="integr15.html#R1" title="INTEGR15:pred.1">is_integrable_on</a> <font color="Olive" title="b2">A</font> &amp;  <a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Olive" title="b3">f</font></span>)</span>,<font color="Olive" title="b2">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<font color="Olive" title="b3">f</font>,<font color="Olive" title="b2">A</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   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><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>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  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="integr15.html#R1" title="INTEGR15: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="integr15.html#V3" title="INTEGR15:attr.3">bounded</a>  holds <br/>( <font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Olive" title="b2">f</font> <a href="integr15.html#R1" title="INTEGR15:pred.1">is_integrable_on</a> <font color="Olive" title="b1">A</font> &amp;  <a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</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="c2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">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="c3">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" 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>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Maroon" title="c3">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="integr15.html#R1" title="INTEGR15:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">A</font> is  <a href="integr15.html#V3" title="INTEGR15:attr.3">bounded</a>  holds <br/>( <font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Olive" title="b1">f</font> <a href="integr15.html#R1" title="INTEGR15:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">A</font> &amp;  <a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Olive" title="b1">f</font></span>)</span>,<font color="Maroon" title="c3">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c3">A</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">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>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">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="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="integr15.html#R1" title="INTEGR15:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">A</font> is  <a href="integr15.html#V3" title="INTEGR15:attr.3">bounded</a>  implies ( <font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c4">f</font> <a href="integr15.html#R1" title="INTEGR15:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">A</font> &amp;  <a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c3">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">A</font>)</span>)</span> ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon" title="c3">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="c4">f</font>
 <span class="kw">and </span><br/><a NAME="E2:43"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="integr15.html#R1" title="INTEGR15:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">and </span><br/><a NAME="E3:43"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">A</font> is  <a href="integr15.html#V3" title="INTEGR15:attr.3">bounded</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c4">f</font> <a href="integr15.html#R1" title="INTEGR15:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">A</font> &amp;  <a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c3">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">A</font>)</span>)</span> )</span><br/>

<div><span class="lab"><font color="Green" title="E37">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">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"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span><br/><a NAME="E1:43_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E2:43_1"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">f</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="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 ;<br/><span class="kw">hence </span><a NAME="E3:43_1"/>
<font color="Maroon" title="c3">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"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="relat_1.html#T27" title="RELAT_1:th.27">RELAT_1:27</a></span>; <a class="txt" 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>
<a NAME="E5:43"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/> <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c3">A</font>)</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> implies  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c3">A</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_2"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c3">A</font>)</span>)</span></span><br/>

<a NAME="E2:43_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">A</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="integr15.html#E28"><span class="lab"><font color="Green" title="E24">Lm6</font></span></a></span>;<br/>
<a NAME="E3:43_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A7</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">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="#E3:43"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E1:43_2"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="ref" href="integr15.html#D15" target="_self" title="INTEGR15:def.15">Def15</a></span>;<br/>
<a NAME="E4:43_2"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
<font color="Maroon" title="c3">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"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E37">A4</font></span></a></span>;<br/>
<a NAME="E5:43_2"/>
<span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E1:43_2"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="ref" href="integr15.html#D16" target="_self" title="INTEGR15:def.16">Def16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:43_2"/>
 <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c3">A</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E4:43_2"><span class="lab"><font color="Green" title="E41">A8</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:43"/><span class="lab"><font color="Green" title="E39">A9</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">A</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c3">A</font>)</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">A</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c3">A</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_3"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">A</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c3">A</font>)</span>)</span></span><br/>

<a NAME="E2:43_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">A</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c3">A</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr15.html#D17" target="_self" title="INTEGR15:def.17">Def17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:43_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">A</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c3">A</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T45" title="RVSUM_1:th.45">RVSUM_1:45</a></span>; <a class="txt" 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>
<a NAME="E7:43"/><span class="lab"><font color="Green" title="E40">A10</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c3">A</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">A</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c3">A</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">A</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font> )</span><br/>

<a NAME="E1:43_4"/><span class="lab"><font color="Green" title="E41">A11</font></span>: 
 <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">A</font>) <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"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">A</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr15.html#T16" target="_self" title="INTEGR15:th.16">Th16</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:43_4"/><span class="lab"><font color="Green" title="E42">A12</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c3">A</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">A</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span><br/>

<a NAME="E3:43_4"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c3">A</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <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"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">A</font>) &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">A</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c3">A</font>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="ref" href="integr15.html#D17" target="_self" title="INTEGR15:def.17">Def17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:43_4"/>
<span class="p1">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c3">A</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">A</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="txt" href="#E2:43_4"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="txt" href="#E1:43_4"><span class="lab"><font color="Green" title="E41">A11</font></span></a></span>; <a class="txt" 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>
<a NAME="E8:43"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_5"/><span class="lab"><font color="Green" title="E41">A13</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">A</font></span><br/>

<a NAME="E2:43_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">A</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="integr15.html#E28"><span class="lab"><font color="Green" title="E24">Lm6</font></span></a></span>;<br/>
<a NAME="E3:43_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A14</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">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="#E3:43"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E1:43_5"><span class="lab"><font color="Green" title="E41">A13</font></span></a>, <a class="ref" href="integr15.html#D15" target="_self" title="INTEGR15:def.15">Def15</a></span>;<br/>
<a NAME="E4:43_5"/><span class="lab"><font color="Green" title="E43">A15</font></span>: 
<font color="Maroon" title="c3">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"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E37">A4</font></span></a></span>;<br/>
<a NAME="E5:43_5"/>
<span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E1:43_5"><span class="lab"><font color="Green" title="E41">A13</font></span></a>, <a class="ref" href="integr15.html#D16" target="_self" title="INTEGR15:def.16">Def16</a></span>;<br/>
<a NAME="E6:43_5"/><span class="kw">then </span>
<font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_5"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E4:43_5"><span class="lab"><font color="Green" title="E43">A15</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="E7:43_5"/>
<span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integr15.html#T16" target="_self" title="INTEGR15:th.16">Th16</a></span>; <a class="txt" 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="E9:43"/>
<font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c4">f</font> <a href="integr15.html#R1" title="INTEGR15:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integr15.html#D16" target="_self" title="INTEGR15:def.16">Def16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c3">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">A</font>)</span>)</span></span><br/>

<a NAME="E10:43"/><span class="lab"><font color="Green" title="E41">A16</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c3">A</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T89" title="FINSEQ_1:th.89">FINSEQ_1:89</a></span>;<br/>
<a NAME="E11:43"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c3">A</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">A</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T89" title="FINSEQ_1:th.89">FINSEQ_1:89</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:43"/>
 <a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="integr15.html#K9" title="INTEGR15:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c3">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><a href="integr15.html#K11" title="INTEGR15:func.11">integral</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">A</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E41">A16</font></span></a>, <a class="ref" href="partfun1.html#T5" title="PARTFUN1:th.5">PARTFUN1:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
