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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b3">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><br/>  for <font color="Olive" title="b4">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> is <font color="Olive" title="b4">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  &amp; <font color="Olive" title="b3">f</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  holds <br/> <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Olive" title="b2">M</font></span>)</span>,<font color="Olive" title="b3">f</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b2">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><br/>  for <font color="Olive" title="b3">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b3">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> is <font color="Olive" title="b3">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  &amp; <font color="Olive" title="b2">f</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  holds <br/> <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Olive" title="b1">M</font></span>)</span>,<font color="Olive" title="b2">f</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><br/>  for <font color="Olive" title="b2">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b2">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> is <font color="Olive" title="b2">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  holds <br/> <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font></span>)</span>,<font color="Olive" title="b1">f</font>)</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 <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> is <font color="Olive" title="b1">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  &amp; <font color="Maroon" title="c4">f</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  holds <br/> <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font></span>)</span>,<font color="Maroon" title="c4">f</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c5">E</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> is <font color="Maroon" title="c5">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  &amp; <font color="Maroon" title="c4">f</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  implies  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font></span>)</span>,<font color="Maroon" title="c4">f</font>) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
<font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="E39">A2</font></span>: 
<font color="Maroon" title="c4">f</font> is <font color="Maroon" title="c5">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 <span class="kw">and </span><br/><a NAME="E3:43"/><span class="lab"><font color="Green" title="E40">A3</font></span>: 
<font color="Maroon" title="c4">f</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font></span>)</span>,<font color="Maroon" title="c4">f</font>)</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">F</font> being   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><span class="kw"> such that </span><br/><a NAME="E5:43"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="E6:43"/><span class="lab"><font color="Green" title="E42">A5</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 <span class="kw">and </span><br/><a NAME="E7:43"/><span class="lab"><font color="Green" title="E43">A6</font></span>: 
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> for <font color="Olive" title="b3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">x</font>
 <span class="kw">and </span><br/><a NAME="E8:43"/><span class="lab"><font color="Green" title="E44">A7</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> holds <br/>( <font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="ref" href="mesfunc5.html#T64" onmouseover="rs('mesfunc5/T64')" onmouseout="rh()">MESFUNC5:64</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">g</font> = <font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/>
<a NAME="E10:43"/><span class="lab"><font color="Green" title="E45">A8</font></span>: 
<font color="Maroon" title="c7">g</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E42">A5</font></span></a></span>;<br/>
<a NAME="E11:43"/><span class="lab"><font color="Green" title="E46">A9</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E41">A4</font></span></a></span>;<br/>
<a NAME="E12:43"/><span class="lab"><font color="Green" title="E47">A10</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">g</font> holds <br/>( <font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp; <font color="Maroon" title="c7">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">g</font> implies ( <font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp; <font color="Maroon" title="c7">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_1"/><span class="lab"><font color="Green" title="E48">A11</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp; <font color="Maroon" title="c7">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> )</span><br/>

<span class="kw">hence </span><a NAME="E2:43_1"/>
<font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="txt" href="#E11:43"><span class="lab"><font color="Green" title="E46">A9</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span></span><br/>

<div><span class="lab"><font color="Green" title="E49">A12</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">m</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">n</font>, <font color="Maroon" title="c10">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">m</font> )</span><br/><span class="kw">assume </span><a NAME="E1:43_1_1"/><span class="lab"><font color="Green" title="E50">A13</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">m</font></span><br/><a NAME="E2:43_1_1"/><span class="lab"><font color="Green" title="E51">A14</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D13" onmouseover="rs('mesfunc5/D13')" onmouseout="rh()">MESFUNC5:def 13</a></span>;<br/><a NAME="E3:43_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D13" onmouseover="rs('mesfunc5/D13')" onmouseout="rh()">MESFUNC5:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:43_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E1:43_1_1"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="txt" href="#E11:43"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="txt" href="#E2:43_1_1"><span class="lab"><font color="Green" title="E51">A14</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:43_1"/><span class="lab"><font color="Green" title="E50">A15</font></span>: 
<font color="Maroon" title="c7">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D13" onmouseover="rs('mesfunc5/D13')" onmouseout="rh()">MESFUNC5:def 13</a></span>;<br/>
<a NAME="E5:43_1"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K12" title="SUPINF_2:func.12">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="ref" href="mesfunc5.html#T54" onmouseover="rs('mesfunc5/T54')" onmouseout="rh()">MESFUNC5:54</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:43_1"/>
<font color="Maroon" title="c7">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1"><span class="lab"><font color="Green" title="E50">A15</font></span></a>, <a class="ref" href="mesfunc5.html#T56" onmouseover="rs('mesfunc5/T56')" onmouseout="rh()">MESFUNC5:56</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c8">K</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a><span class="kw"> such that </span><br/><a NAME="E14:43"/><span class="lab"><font color="Green" title="E48">A16</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c8">K</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>)
 <span class="kw">and </span><br/><a NAME="E15:43"/><span class="lab"><font color="Green" title="E49">A17</font></span>: 
<font color="Maroon" title="c8">K</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E16:43"/>
 <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K12" title="SUPINF_2:func.12">rng</a> <font color="Maroon" title="c8">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c8">K</font>
 <span class="kw">and </span><br/><a NAME="E17:43"/>
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">g</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c8">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E11:43"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E12:43"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="mesfunc5.html#T75" onmouseover="rs('mesfunc5/T75')" onmouseout="rh()">MESFUNC5:75</a></span>;<br/>
<a NAME="E18:43"/><span class="lab"><font color="Green" title="E50">A18</font></span>: 
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c8">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="txt" href="#E14:43"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E15:43"><span class="lab"><font color="Green" title="E49">A17</font></span></a>, <a class="ref" href="mesfunc5.html#D15" onmouseover="rs('mesfunc5/D15')" onmouseout="rh()">MESFUNC5:def 15</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">E1</font> = <font color="Maroon" title="c5">E</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>) <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun14.html#T27" target="_self" onmouseover="rs('mesfun14/T27')" onmouseout="rh()">Th27</a></span>;<br/>
<a NAME="E20:43"/><span class="lab"><font color="Green" title="E51">A19</font></span>: 
<font color="Maroon" title="c4">f</font> is <font color="Maroon" title="c9">E1</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="mesfun14.html#T30" target="_self" onmouseover="rs('mesfun14/T30')" onmouseout="rh()">Th30</a></span>;<br/>
<a NAME="E21:43"/><span class="lab"><font color="Green" title="E52">A20</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a>  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>) &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="ref" href="mesfun14.html#T33" target="_self" onmouseover="rs('mesfun14/T33')" onmouseout="rh()">Th33</a></span>;<br/>
<a NAME="E22:43"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c8">K</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" 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="c10">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">K</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>)</span><br/>

<a NAME="E1:43_2"/><span class="lab"><font color="Green" title="E53">A21</font></span>: 
<font color="Maroon" title="c8">K</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:43"><span class="lab"><font color="Green" title="E48">A16</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:43_2_1"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_2_1_1"/><span class="lab"><font color="Green" title="E54">A22</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">K</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>)</span><br/><div class="add"><a NAME="E2:43_2_1_1"/>
( <font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c10">n</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E42">A5</font></span></a></span>;<br/><a NAME="E3:43_2_1_1"/><span class="kw">then </span>
 <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun14.html#T36" target="_self" onmouseover="rs('mesfun14/T36')" onmouseout="rh()">Th36</a></span>;<br/><a NAME="E4:43_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">K</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2_1_1"><span class="lab"><font color="Green" title="E54">A22</font></span></a>, <a class="txt" href="#E1:43_2"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="ref" href="mesfunc5.html#D14" onmouseover="rs('mesfunc5/D14')" onmouseout="rh()">MESFUNC5:def 14</a></span>;<br/><span class="kw">hence </span><a NAME="E5:43_2_1_1"/>
<font color="Maroon" title="c8">K</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2_1_1"><span class="lab"><font color="Green" title="E54">A22</font></span></a>, <a class="ref" href="mesfunc5.html#D14" onmouseover="rs('mesfunc5/D14')" onmouseout="rh()">MESFUNC5:def 14</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_2_1_2"/><span class="lab"><font color="Green" title="E54">A23</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">K</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>)</span><br/><div class="add"><a NAME="E2:43_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">K</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="ref" href="mesfunc5.html#D14" onmouseover="rs('mesfunc5/D14')" onmouseout="rh()">MESFUNC5:def 14</a></span>;<br/><span class="kw">hence </span><a NAME="E3:43_2_1_2"/>
<font color="Maroon" title="c8">K</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2_1_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="ref" href="mesfunc5.html#D14" onmouseover="rs('mesfunc5/D14')" onmouseout="rh()">MESFUNC5:def 14</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E23:43"/>
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <font color="Maroon" title="c3">M</font></span>)</span>,<font color="Maroon" title="c4">f</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:43"><span class="lab"><font color="Green" title="E50">A18</font></span></a>, <a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E20:43"><span class="lab"><font color="Green" title="E51">A19</font></span></a>, <a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E21:43"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="txt" href="#E15:43"><span class="lab"><font color="Green" title="E49">A17</font></span></a>, <a class="ref" href="mesfunc5.html#D15" onmouseover="rs('mesfunc5/D15')" onmouseout="rh()">MESFUNC5:def 15</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
