<?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> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">M</font> be    <a href="measure1.html#M3" title="MEASURE1:mode.3">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/>







<b>assume </b><a NAME="E1:118"/><i><font color="Green" title="E112">A1</font></i>: 
(  ex <font color="Olive" title="b1">A</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">A</font> <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="c4">f</font> &amp; <font color="Olive" title="b1">A</font> <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="c5">g</font> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Olive" title="b1">A</font> ) &amp; <font color="Maroon" title="c4">f</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <font color="Maroon" title="c5">g</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</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>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font> holds <br/><font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</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="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> ) )
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E3:118"/><i><font color="Green" title="E113">A2</font></i>: 
( <font color="Maroon" title="c6">A</font> <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="c4">f</font> &amp; <font color="Maroon" title="c6">A</font> <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="c5">g</font> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c6">A</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:118"><i><font color="Green" title="E112">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">F</font> being    <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> , <font color="Maroon" title="c8">K</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E5:118"/><i><font color="Green" title="E114">A3</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 <br/> ( <font color="Maroon" title="c7">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc2.html#R2" title="MESFUNC2:pred.2">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> ) ) &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="c7">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> ) &amp; ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/>for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b3">x</font> <a href="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="c4">f</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b3">x</font> ) &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>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> holds <br/>( <font color="Maroon" title="c7">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> ) ) &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="c8">K</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) &amp; <font color="Maroon" title="c8">K</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c8">K</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:118"><i><font color="Green" title="E112">A1</font></i></a>, <a class="ref" href="mesfunc5.html#D15" target="_self" title="MESFUNC5:def.15">Def15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">G</font> being    <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> , <font color="Maroon" title="c10">L</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E7:118"/><i><font color="Green" title="E115">A4</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 <br/> ( <font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc2.html#R2" title="MESFUNC2:pred.2">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font> ) ) &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="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> ) &amp; ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/>for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b3">x</font> <a href="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="c5">g</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b3">x</font> ) &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>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font> holds <br/>( <font color="Maroon" title="c9">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> ) ) &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="c10">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) &amp; <font color="Maroon" title="c10">L</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c10">L</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:118"><i><font color="Green" title="E112">A1</font></i></a>, <a class="ref" href="mesfunc5.html#D15" target="_self" title="MESFUNC5:def.15">Def15</a></i>;<br/>
<a NAME="E8:118"/><i><font color="Green" title="E116">A5</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="c6">A</font> holds <br/> <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E1:118_1"/><i><font color="Green" title="E117">A6</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">n</font>, <font color="Maroon" title="c13">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:118_1_1"/><i><font color="Green" title="E118">A7</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font>
 ;<br/><a NAME="E2:118_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#D13" target="_self" title="MESFUNC5:def.13">Def13</a></i>;<br/><b>hence </b><a NAME="E3:118_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c13">m</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:118_1"><i><font color="Green" title="E117">A6</font></i></a>, <a class="txt" href="mesfunc5.html#E3:118"><i><font color="Green" title="E113">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E7:118"><i><font color="Green" title="E115">A4</font></i></a>, <a class="txt" href="mesfunc5.html#E1:118_1_1"><i><font color="Green" title="E118">A7</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:118_1"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T60" target="_self" title="MESFUNC5:th.60">Th60</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:118"/><i><font color="Green" title="E117">A8</font></i>: 
for <font color="Olive" title="b1">n0</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <font color="Maroon" title="c8">K</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c8">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c8">K</font> &amp; <font color="Maroon" title="c10">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c8">K</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">n0</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>set </b><font color="Maroon" title="c12">gg</font> = <font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n0</font>;<br/>
<a NAME="E1:118_2"/><i><font color="Green" title="E118">A9</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n0</font> <a href="mesfunc2.html#R2" title="MESFUNC2:pred.2">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n0</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E7:118"><i><font color="Green" title="E115">A4</font></i></a>, <a class="txt" href="mesfunc5.html#E3:118"><i><font color="Green" title="E113">A2</font></i></a></i>;<br/>
<a NAME="E2:118_2"/>
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"><font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n0</font></span>)</span> holds <br/>( <font color="Maroon" title="c7">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n0</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E1:118_2_1"/><i><font color="Green" title="E119">A10</font></i>: 
<font color="Maroon" title="c13">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="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n0</font></span>)</span>
 ;<br/>

<a NAME="E2:118_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n0</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">n0</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D13" target="_self" title="MESFUNC5:def.13">Def13</a>, <a class="ref" href="mesfunc5.html#T62" target="_self" title="MESFUNC5:th.62">Th62</a></i>;<br/>
<a NAME="E3:118_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n0</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c13">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c13">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c13">x</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:118"><i><font color="Green" title="E112">A1</font></i></a>, <a class="txt" href="mesfunc5.html#E3:118"><i><font color="Green" title="E113">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E8:118"><i><font color="Green" title="E116">A5</font></i></a>, <a class="txt" href="mesfunc5.html#E1:118_2_1"><i><font color="Green" title="E119">A10</font></i></a>, <a class="txt" href="mesfunc5.html#E1:118_2"><i><font color="Green" title="E118">A9</font></i></a></i>;<br/>
<a NAME="E4:118_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n0</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c13">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c13">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E5:118"><i><font color="Green" title="E114">A3</font></i></a>, <a class="txt" href="mesfunc5.html#E7:118"><i><font color="Green" title="E115">A4</font></i></a>, <a class="txt" href="mesfunc5.html#E1:118_2_1"><i><font color="Green" title="E119">A10</font></i></a>, <a class="txt" href="mesfunc5.html#E1:118_2"><i><font color="Green" title="E118">A9</font></i></a>, <a class="txt" href="mesfunc5.html#E3:118"><i><font color="Green" title="E113">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:118_2_1"/>
( <font color="Maroon" title="c7">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n0</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c13">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="mesfunc5.html#E5:118"><i><font color="Green" title="E114">A3</font></i></a>, <a class="txt" href="mesfunc5.html#E3:118"><i><font color="Green" title="E113">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E1:118_2_1"><i><font color="Green" title="E119">A10</font></i></a>, <a class="txt" href="mesfunc5.html#E1:118_2"><i><font color="Green" title="E118">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c13">GG</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E4:118_2"/><i><font color="Green" title="E119">A11</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="c13">GG</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) &amp; <font color="Maroon" title="c13">GG</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c13">GG</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c13">GG</font> &amp;  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n0</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c13">GG</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T81" target="_self" title="MESFUNC5:th.81">Th81</a>, <a class="txt" href="mesfunc5.html#E5:118"><i><font color="Green" title="E114">A3</font></i></a>, <a class="txt" href="mesfunc5.html#E3:118"><i><font color="Green" title="E113">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E1:118_2"><i><font color="Green" title="E118">A9</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:118_2_2"/>
<font color="Maroon" title="c13">GG</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:118_2"><i><font color="Green" title="E119">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E2:118_2_2"/>
<font color="Maroon" title="c13">GG</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">K</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c14">n</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:118"><i><font color="Green" title="E114">A3</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:118_2"/><b>then </b>
<font color="Maroon" title="c13">GG</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">K</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>
<b>hence </b><a NAME="E7:118_2"/>
( <font color="Maroon" title="c8">K</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c8">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c8">K</font> &amp; <font color="Maroon" title="c10">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c8">K</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E7:118"><i><font color="Green" title="E115">A4</font></i></a>, <a class="txt" href="mesfunc5.html#E4:118_2"><i><font color="Green" title="E119">A11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:118"/>
for <font color="Olive" title="b1">n0</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <font color="Maroon" title="c10">L</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c10">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c10">L</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">n0</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>set </b><font color="Maroon" title="c12">ff</font> = <font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n0</font>;<br/>
<a NAME="E1:118_3"/><i><font color="Green" title="E118">A12</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n0</font> <a href="mesfunc2.html#R2" title="MESFUNC2:pred.2">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n0</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E7:118"><i><font color="Green" title="E115">A4</font></i></a>, <a class="txt" href="mesfunc5.html#E3:118"><i><font color="Green" title="E113">A2</font></i></a></i>;<br/>
<a NAME="E2:118_3"/>
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"><font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n0</font></span>)</span> holds <br/>( <font color="Maroon" title="c9">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n0</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E1:118_3_1"/><i><font color="Green" title="E119">A13</font></i>: 
<font color="Maroon" title="c13">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="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n0</font></span>)</span>
 ;<br/>

<a NAME="E2:118_3_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n0</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">n0</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D13" target="_self" title="MESFUNC5:def.13">Def13</a>, <a class="ref" href="mesfunc5.html#T62" target="_self" title="MESFUNC5:th.62">Th62</a></i>;<br/>
<b>hence </b><a NAME="E3:118_3_1"/>
( <font color="Maroon" title="c9">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n0</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c13">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E7:118"><i><font color="Green" title="E115">A4</font></i></a>, <a class="txt" href="mesfunc5.html#E3:118"><i><font color="Green" title="E113">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E8:118"><i><font color="Green" title="E116">A5</font></i></a>, <a class="txt" href="mesfunc5.html#E1:118_3"><i><font color="Green" title="E118">A12</font></i></a>, <a class="txt" href="mesfunc5.html#E1:118_3_1"><i><font color="Green" title="E119">A13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c13">FF</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E4:118_3"/><i><font color="Green" title="E119">A14</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="c13">FF</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) &amp; <font color="Maroon" title="c13">FF</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c13">FF</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c13">FF</font> &amp;  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c11">n0</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c13">FF</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T81" target="_self" title="MESFUNC5:th.81">Th81</a>, <a class="txt" href="mesfunc5.html#E7:118"><i><font color="Green" title="E115">A4</font></i></a>, <a class="txt" href="mesfunc5.html#E3:118"><i><font color="Green" title="E113">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E1:118_3"><i><font color="Green" title="E118">A12</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>thus </b><font color="Maroon" title="c13">FF</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c14">n</font> = 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:118_3"><i><font color="Green" title="E119">A14</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">L</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c14">n</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E7:118"><i><font color="Green" title="E115">A4</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E6:118_3"/><b>then </b>
<font color="Maroon" title="c13">FF</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c10">L</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>
<b>hence </b><a NAME="E7:118_3"/>
( <font color="Maroon" title="c10">L</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <font color="Maroon" title="c10">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c10">L</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:118_3"><i><font color="Green" title="E119">A14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:118"/>
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">g</font> <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="c3">M</font>,<font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:118"><i><font color="Green" title="E114">A3</font></i></a>, <a class="txt" href="mesfunc5.html#E7:118"><i><font color="Green" title="E115">A4</font></i></a>, <a class="txt" href="mesfunc5.html#E9:118"><i><font color="Green" title="E117">A8</font></i></a>, <a class="ref" href="mesfunc5.html#T63" target="_self" title="MESFUNC5:th.63">Th63</a></i>;<br/>


</div>
