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

<b>let </b><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#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <br/> for <font color="Olive" title="b2">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="b3">M</font> being  <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b2">S</font><br/> for <font color="Olive" title="b4">E</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b2">S</font>  st <font color="Olive" title="b3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b4">E</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Olive" title="b4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; ( for <font color="Olive" title="b5">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b5">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b4">E</font> ) &amp; <font color="Olive" title="b1">F</font> is <a href="mesfun10.html#V1" title="MESFUN10:attr.1">uniformly_bounded</a> &amp; ( for <font color="Olive" title="b5">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="b5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">E</font> holds <br/><font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b5">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> ) holds <br/>( ( for <font color="Olive" title="b5">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b5">n</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Olive" title="b3">M</font> ) &amp;  <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Olive" title="b1">F</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Olive" title="b3">M</font> &amp;  ex <font color="Olive" title="b5">I</font> being  <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( ( for <font color="Olive" title="b6">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b5">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Olive" title="b3">M</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b6">n</font></span>)</span> ) &amp; <font color="Olive" title="b5">I</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> <font color="Olive" title="b5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Olive" title="b3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Olive" title="b1">F</font></span>)</span> ) )</span><br/><b>let </b><font color="Maroon" title="c2">F</font> be  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">E</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b2">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b3">E</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Olive" title="b3">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; ( for <font color="Olive" title="b4">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b3">E</font> ) &amp; <font color="Maroon" title="c2">F</font> is <a href="mesfun10.html#V1" title="MESFUN10:attr.1">uniformly_bounded</a> &amp; ( for <font color="Olive" title="b4">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="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">E</font> holds <br/><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b4">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> ) holds <br/>( ( for <font color="Olive" title="b4">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">n</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Olive" title="b2">M</font> ) &amp;  <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Olive" title="b2">M</font> &amp;  ex <font color="Olive" title="b4">I</font> being  <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( ( for <font color="Olive" title="b5">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b4">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Olive" title="b2">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b5">n</font></span>)</span> ) &amp; <font color="Olive" title="b4">I</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> <font color="Olive" title="b4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Olive" title="b2">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> ) )</span><br/><b>let </b><font color="Maroon" title="c3">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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c3">S</font><br/> for <font color="Olive" title="b2">E</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">S</font>  st <font color="Olive" title="b1">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">E</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Olive" title="b2">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; ( for <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b2">E</font> ) &amp; <font color="Maroon" title="c2">F</font> is <a href="mesfun10.html#V1" title="MESFUN10:attr.1">uniformly_bounded</a> &amp; ( 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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">E</font> holds <br/><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b3">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> ) holds <br/>( ( for <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">n</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Olive" title="b1">M</font> ) &amp;  <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Olive" title="b1">M</font> &amp;  ex <font color="Olive" title="b3">I</font> being  <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( ( for <font color="Olive" title="b4">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b3">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Olive" title="b1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">n</font></span>)</span> ) &amp; <font color="Olive" title="b3">I</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> <font color="Olive" title="b3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Olive" title="b1">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> ) )</span><br/><b>let </b><font color="Maroon" title="c4">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">E</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">S</font>  st <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">E</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Olive" title="b1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font> ) &amp; <font color="Maroon" title="c2">F</font> is <a href="mesfun10.html#V1" title="MESFUN10:attr.1">uniformly_bounded</a> &amp; ( for <font color="Olive" title="b2">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="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> holds <br/><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b2">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> ) holds <br/>( ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> ) &amp;  <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> &amp;  ex <font color="Olive" title="b2">I</font> being  <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( ( for <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b2">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">n</font></span>)</span> ) &amp; <font color="Olive" title="b2">I</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> <font color="Olive" title="b2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> ) )</span><br/><b>let </b><font color="Maroon" title="c5">E</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">E</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font> ) &amp; <font color="Maroon" title="c2">F</font> is <a href="mesfun10.html#V1" title="MESFUN10:attr.1">uniformly_bounded</a> &amp; ( 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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> holds <br/><font color="Maroon" title="c2">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> ) implies ( ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> ) &amp;  <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> &amp;  ex <font color="Olive" title="b1">I</font> being  <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span> ) &amp; <font color="Olive" title="b1">I</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> <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> ) ) )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:21"/><i><font color="Green" title="E20">A1</font></i>: 
( <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">E</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> )
 <b>and </b><br/><a NAME="E2:21"/><i><font color="Green" title="E21">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font>
 <b>and </b><br/><a NAME="E3:21"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="mesfun10.html#V1" title="MESFUN10:attr.1">uniformly_bounded</a>
 <b>and </b><br/><a NAME="E4:21"/><i><font color="Green" title="E23">A4</font></i>: 
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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> holds <br/><font color="Maroon" title="c2">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> ) &amp;  <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> &amp;  ex <font color="Olive" title="b1">I</font> being  <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span> ) &amp; <font color="Olive" title="b1">I</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> <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> ) )</span><br/>

<b>consider </b><font color="Maroon" title="c6">K1</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E6:21"/><i><font color="Green" title="E24">A5</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> holds <br/><span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">K1</font>
 <b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="mesfun10.html#D1" target="_self" title="MESFUN10:def.1">DefUB</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">K</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c6">K1</font>,1;<br/>
<a NAME="E7:21"/><i><font color="Green" title="E25">X</font></i>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c6">K1</font>,1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T30" title="XXREAL_0:th.30">XXREAL_0:30</a></i>;<br/>
<a NAME="E8:21"/><i><font color="Green" title="E26">Y</font></i>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c6">K1</font>,1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">K1</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<a NAME="E9:21"/><i><font color="Green" title="E27">X5</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> holds <br/><span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c6">K1</font>,1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> holds <br/><span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c6">K1</font>,1</span><br/><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> implies <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c6">K1</font>,1 )</span><br/>



<b>assume </b><a NAME="E1:21_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c6">K1</font>,1</span><br/>

<a NAME="E2:21_1"/><b>then </b>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">K1</font>
 
<b>by </b><i><a class="txt" href="#E6:21"><i><font color="Green" title="E24">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:21_1"/>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c6">K1</font>,1
 <b>by </b><i><a class="txt" href="#E8:21"><i><font color="Green" title="E26">Y</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E10:21"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c6">K1</font>,1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">h</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#K7" title="NUMBERS:func.7">ExtREAL</a> <b> such that </b><br/><a NAME="E12:21"/><i><font color="Green" title="E28">B1</font></i>: 
( <font color="Maroon" title="c8">h</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c3">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> holds <br/><font color="Maroon" title="c8">h</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c6">K1</font>,1 ) )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T47" title="MESFUNC5:th.47">MESFUNC5:47</a>, <a class="ref" href="numbers.html#T31" title="NUMBERS:th.31">NUMBERS:31</a></i>;<br/>
<a NAME="E13:21"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> holds <br/><font color="Maroon" title="c8">h</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<b>by </b><i><a class="txt" href="#E12:21"><i><font color="Green" title="E28">B1</font></i></a>, <a class="txt" href="#E7:21"><i><font color="Green" title="E25">X</font></i></a></i>;<br/>
<a NAME="E14:21"/><b>then </b><i><font color="Green" title="E29">B2</font></i>: 
<font color="Maroon" title="c8">h</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="#E12:21"><i><font color="Green" title="E28">B1</font></i></a>, <a class="ref" href="supinf_2.html#T71" title="SUPINF_2:th.71">SUPINF_2:71</a></i>;<br/>
<a NAME="E15:21"/><i><font color="Green" title="E30">B3</font></i>: 
<font color="Maroon" title="c8">h</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font>
 
<b>by </b><i><a class="txt" href="#E12:21"><i><font color="Green" title="E28">B1</font></i></a>, <a class="ref" href="mesfunc2.html#T37" title="MESFUNC2:th.37">MESFUNC2:37</a></i>;<br/>
<a NAME="E16:21"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c6">K1</font>,1 is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E17:21"/><b>then </b><i><font color="Green" title="E31">B4</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c6">K1</font>,1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">h</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E12:21"><i><font color="Green" title="E28">B1</font></i></a>, <a class="txt" href="#E7:21"><i><font color="Green" title="E25">X</font></i></a>, <a class="ref" href="mesfunc5.html#T110" title="MESFUNC5:th.110">MESFUNC5:110</a></i>;<br/>
<a NAME="E18:21"/><i><font color="Green" title="E32">B5</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c6">K1</font>,1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="#E7:21"><i><font color="Green" title="E25">X</font></i></a>, <a class="ref" href="extreal1.html#D1" title="EXTREAL1:def.1">EXTREAL1:def 1</a></i>;<br/>
<a NAME="E19:21"/><i><font color="Green" title="E33">B6</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c8">h</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<a NAME="E20:21"/><i><font color="Green" title="E34">B7</font></i>: 
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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">h</font> holds <br/><font color="Maroon" title="c8">h</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c8">h</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">h</font> implies <font color="Maroon" title="c8">h</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c8">h</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font> )</span><br/>

<b>assume </b><a NAME="E1:21_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">h</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c8">h</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font></span><br/>

<a NAME="E2:21_2"/><b>then </b><i><font color="Green" title="E35">B71</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c8">h</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<a NAME="E3:21_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">h</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font>
 
<b>by </b><i><a class="txt" href="#E14:21"><i><font color="Green" title="E29">B2</font></i></a>, <a class="ref" href="supinf_2.html#T70" title="SUPINF_2:th.70">SUPINF_2:70</a></i>;<br/>
<a NAME="E4:21_2"/><b>then </b>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font>
 
<b>by </b><i><a class="ref" href="extreal1.html#D3" title="EXTREAL1:def.3">EXTREAL1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:21_2"/>
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c8">h</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="#E2:21_2"><i><font color="Green" title="E35">B71</font></i></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E21:21"/><i><font color="Green" title="E35">B8</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">h</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">h</font> )
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D2" title="MESFUNC2:def.2">MESFUNC2:def 2</a>, <a class="ref" href="mesfunc2.html#D3" title="MESFUNC2:def.3">MESFUNC2:def 3</a></i>;<br/>
<a NAME="E22:21"/><i><font color="Green" title="E36">B9</font></i>: 
(  <a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c8">h</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font> &amp;  <a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c8">h</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font> )
 
<b>by </b><i><a class="txt" href="#E12:21"><i><font color="Green" title="E28">B1</font></i></a>, <a class="txt" href="#E15:21"><i><font color="Green" title="E30">B3</font></i></a>, <a class="ref" href="mesfunc2.html#T27" title="MESFUNC2:th.27">MESFUNC2:27</a>, <a class="ref" href="mesfunc2.html#T28" title="MESFUNC2:th.28">MESFUNC2:28</a></i>;<br/>
<a NAME="E23:21"/>
( ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c8">h</font></span>)</span> holds <br/> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> ) &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c8">h</font></span>)</span> holds <br/> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> ) )
 
<b>by </b><i><a class="ref" href="mesfunc2.html#T14" title="MESFUNC2:th.14">MESFUNC2:14</a>, <a class="ref" href="mesfunc2.html#T15" title="MESFUNC2:th.15">MESFUNC2:15</a></i>;<br/>
<a NAME="E24:21"/><b>then </b><i><font color="Green" title="E37">C1</font></i>: 
(  <a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c8">h</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp;  <a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c8">h</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 
<b>by </b><i><a class="ref" href="supinf_2.html#T71" title="SUPINF_2:th.71">SUPINF_2:71</a></i>;<br/>
<a NAME="E25:21"/><b>then </b><i><font color="Green" title="E38">C2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c8">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c8">h</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c8">h</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T28" title="MESFUNC5:th.28">MESFUNC5:28</a></i>;<br/>
<a NAME="E26:21"/><i><font color="Green" title="E39">C3</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c8">h</font></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c8">h</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E12:21"><i><font color="Green" title="E28">B1</font></i></a>, <a class="txt" href="#E21:21"><i><font color="Green" title="E35">B8</font></i></a>, <a class="txt" href="#E22:21"><i><font color="Green" title="E36">B9</font></i></a>, <a class="txt" href="#E24:21"><i><font color="Green" title="E37">C1</font></i></a>, <a class="ref" href="mesfunc5.html#T85" title="MESFUNC5:th.85">MESFUNC5:85</a></i>;<br/>
<a NAME="E27:21"/><i><font color="Green" title="E40">C4</font></i>: 
 ex <font color="Olive" title="b1">C</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c8">h</font></span>)</span></span>)</span> &amp;  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c8">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c4">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">C</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c4">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">C</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E12:21"><i><font color="Green" title="E28">B1</font></i></a>, <a class="txt" href="#E21:21"><i><font color="Green" title="E35">B8</font></i></a>, <a class="txt" href="#E22:21"><i><font color="Green" title="E36">B9</font></i></a>, <a class="txt" href="#E24:21"><i><font color="Green" title="E37">C1</font></i></a>, <a class="ref" href="mesfunc5.html#T84" title="MESFUNC5:th.84">MESFUNC5:84</a></i>;<br/>
<a NAME="E28:21"/><i><font color="Green" title="E41">C5</font></i>: 
( <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c8">h</font> &amp; <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c8">h</font> )
 
<b>by </b><i><a class="txt" href="#E12:21"><i><font color="Green" title="E28">B1</font></i></a>, <a class="txt" href="#E21:21"><i><font color="Green" title="E35">B8</font></i></a>, <a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></i>;<br/>
<a NAME="E29:21"/>
(  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c8">h</font> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c8">h</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="c4">M</font>,<font color="Maroon" title="c8">h</font> )
 
<b>by </b><i><a class="txt" href="#E12:21"><i><font color="Green" title="E28">B1</font></i></a>, <a class="txt" href="#E14:21"><i><font color="Green" title="E29">B2</font></i></a>, <a class="ref" href="mesfunc5.html#T95" title="MESFUNC5:th.95">MESFUNC5:95</a></i>;<br/>
<a NAME="E30:21"/><b>then </b>
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c8">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="#E18:21"><i><font color="Green" title="E32">B5</font></i></a>, <a class="txt" href="#E17:21"><i><font color="Green" title="E31">B4</font></i></a>, <a class="txt" href="#E12:21"><i><font color="Green" title="E28">B1</font></i></a>, <a class="txt" href="#E1:21"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E7:21"><i><font color="Green" title="E25">X</font></i></a>, <a class="ref" href="extreal1.html#T44" title="EXTREAL1:th.44">EXTREAL1:44</a></i>;<br/>
<a NAME="E31:21"/><b>then </b>
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c4">M</font>,<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c8">h</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="#E20:21"><i><font color="Green" title="E34">B7</font></i></a>, <a class="txt" href="#E19:21"><i><font color="Green" title="E33">B6</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>
<a NAME="E32:21"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c4">M</font>,<span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c8">h</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c4">M</font>,<span class="p2">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c8">h</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="#E28:21"><i><font color="Green" title="E41">C5</font></i></a>, <a class="txt" href="#E25:21"><i><font color="Green" title="E38">C2</font></i></a>, <a class="txt" href="#E12:21"><i><font color="Green" title="E28">B1</font></i></a>, <a class="txt" href="#E21:21"><i><font color="Green" title="E35">B8</font></i></a>, <a class="txt" href="#E27:21"><i><font color="Green" title="E40">C4</font></i></a>, <a class="ref" href="mesfunc2.html#T26" title="MESFUNC2:th.26">MESFUNC2:26</a></i>;<br/>
<a NAME="E33:21"/><b>then </b>
(  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp;  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="txt" href="#E26:21"><i><font color="Green" title="E39">C3</font></i></a>, <a class="ref" href="extreal2.html#T23" title="EXTREAL2:th.23">EXTREAL2:23</a></i>;<br/>
<a NAME="E34:21"/><b>then </b>
(  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp;  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/>
<a NAME="E35:21"/><b>then </b><i><font color="Green" title="E42">C6</font></i>: 
<font color="Maroon" title="c8">h</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font>
 
<b>by </b><i><a class="txt" href="#E12:21"><i><font color="Green" title="E28">B1</font></i></a>, <a class="txt" href="#E15:21"><i><font color="Green" title="E30">B3</font></i></a>, <a class="ref" href="mesfunc5.html#D17" title="MESFUNC5:def.17">MESFUNC5:def 17</a></i>;<br/>
<a NAME="E36:21"/><i><font color="Green" title="E43">C7</font></i>: 
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><br/> for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> holds <br/><span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">h</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> holds <br/><span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">h</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font></span><br/><b>let </b><font color="Maroon" title="c10">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> implies <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">h</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font> )</span><br/>



<b>assume </b><a NAME="E1:21_3"/><i><font color="Green" title="E44">C71</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">h</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font></span><br/>

<a NAME="E2:21_3"/><b>then </b><i><font color="Green" title="E45">C72</font></i>: 
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c6">K1</font>,1
 
<b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E9:21"><i><font color="Green" title="E27">X5</font></i></a></i>;<br/>
<a NAME="E3:21_3"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<a NAME="E4:21_3"/><b>then </b>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span>
 
<b>by </b><i><a class="txt" href="#E1:21_3"><i><font color="Green" title="E44">C71</font></i></a>, <a class="txt" href="#E1:21"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></i>;<br/>
<a NAME="E5:21_3"/><b>then </b>
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E6:21_3"/>
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">h</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="#E2:21_3"><i><font color="Green" title="E45">C72</font></i></a>, <a class="txt" href="#E1:21_3"><i><font color="Green" title="E44">C71</font></i></a>, <a class="txt" href="#E12:21"><i><font color="Green" title="E28">B1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E37:21"/><b>then </b><i><font color="Green" title="E44">D1</font></i>: 
( ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> ) &amp; <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> &amp; <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> )
 
<b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E2:21"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="#E12:21"><i><font color="Green" title="E28">B1</font></i></a>, <a class="txt" href="#E14:21"><i><font color="Green" title="E29">B2</font></i></a>, <a class="txt" href="#E35:21"><i><font color="Green" title="E42">C6</font></i></a>, <a class="ref" href="mesfun10.html#T17" target="_self" title="MESFUN10:th.17">Th136z</a></i>;<br/>
<div><i><font color="Green" title="E45">E1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">n</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font></span><br/><a NAME="E1:21_4"/>
( <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font> )
 <b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E2:21"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="#E12:21"><i><font color="Green" title="E28">B1</font></i></a>, <a class="txt" href="#E14:21"><i><font color="Green" title="E29">B2</font></i></a>, <a class="txt" href="#E35:21"><i><font color="Green" title="E42">C6</font></i></a>, <a class="txt" href="#E36:21"><i><font color="Green" title="E43">C7</font></i></a>, <a class="ref" href="mesfun10.html#T17" target="_self" title="MESFUN10:th.17">Th136z</a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></i>;<br/><b>hence </b><a NAME="E2:21_4"/>
<font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">n</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T106" title="MESFUNC5:th.106">MESFUNC5:106</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E39:21"/><i><font color="Green" title="E46">E4</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc8.html#D8" title="MESFUNC8:def.8">MESFUNC8:def 8</a>, <a class="ref" href="mesfunc8.html#D10" title="MESFUNC8:def.10">MESFUNC8:def 10</a></i>;<br/>
<a NAME="E40:21"/>
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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> implies <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font> )</span><br/>

<b>assume </b><a NAME="E1:21_5"/><i><font color="Green" title="E47">E41</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font></span><br/>

<a NAME="E2:21_5"/><b>then </b>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font>
 
<b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="mesfunc8.html#D10" title="MESFUNC8:def.10">MESFUNC8:def 10</a></i>;<br/>
<a NAME="E3:21_5"/><b>then </b>
<font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c9">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 
<b>by </b><i><a class="txt" href="#E4:21"><i><font color="Green" title="E23">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:21_5"/>
<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="#E1:21_5"><i><font color="Green" title="E47">E41</font></i></a>, <a class="ref" href="mesfunc8.html#T14" title="MESFUNC8:th.14">MESFUNC8:14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E41:21"/><b>then </b><i><font color="Green" title="E47">E2</font></i>: 
 <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E39:21"><i><font color="Green" title="E46">E4</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>
<a NAME="E42:21"/><b>then </b><i><font color="Green" title="E48">E3</font></i>: 
 <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font>
 
<b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E2:21"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="mesfunc8.html#T24" title="MESFUNC8:th.24">MESFUNC8:24</a></i>;<br/>
<a NAME="E43:21"/>
 ex <font color="Olive" title="b1">I</font> being  <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span> ) &amp;  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Olive" title="b1">I</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span> &amp;  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Olive" title="b1">I</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span> &amp; ( ( for <font color="Olive" title="b2">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="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> holds <br/><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b2">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> ) implies ( <font color="Olive" title="b1">I</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> <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> ) ) )
 
<b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E2:21"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="#E12:21"><i><font color="Green" title="E28">B1</font></i></a>, <a class="txt" href="#E14:21"><i><font color="Green" title="E29">B2</font></i></a>, <a class="txt" href="#E35:21"><i><font color="Green" title="E42">C6</font></i></a>, <a class="txt" href="#E36:21"><i><font color="Green" title="E43">C7</font></i></a>, <a class="ref" href="mesfun10.html#T18" target="_self" title="MESFUN10:th.18">Th136x</a></i>;<br/>
<b>hence </b><a NAME="E44:21"/>
( ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> ) &amp;  <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> &amp;  ex <font color="Olive" title="b1">I</font> being  <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span> ) &amp; <font color="Olive" title="b1">I</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> <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="#E4:21"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="#E38:21"><i><font color="Green" title="E45">E1</font></i></a>, <a class="txt" href="#E41:21"><i><font color="Green" title="E47">E2</font></i></a>, <a class="txt" href="#E42:21"><i><font color="Green" title="E48">E3</font></i></a>, <a class="txt" href="#E1:21"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E39:21"><i><font color="Green" title="E46">E4</font></i></a>, <a class="txt" href="#E37:21"><i><font color="Green" title="E44">D1</font></i></a>, <a class="ref" href="mesfunc5.html#T106" title="MESFUNC5:th.106">MESFUNC5:106</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
