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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">M</font> be    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">F</font> be    <a href="seqfunc.html#M1">Functional_Sequence</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">L</font> be   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:105"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">g</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E2:105"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/>0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 <b>and </b><br/><a NAME="E3:105"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E4:105"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds   <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E5:105"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>and </b><br/><a NAME="E6:105"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 <b>and </b><br/><a NAME="E7:105"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/>( <font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">x</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp; <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">x</font></span>)</span> )
 <b>and </b><br/><a NAME="E8:105"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1"><b>suppose </b></a><a NAME="E1:105_1_1"/><i><font color="Green">E44</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:105_1_1_1"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E2:105_1_1_1"/><b>then </b>
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E3:105_1_1_1"/>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> 0
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E3:105_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">L</font> is <a href="mesfunc5.html#V7" target="_self">convergent_to_finite_number</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> 0 )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:105_1_1"/>
<font color="Maroon">L</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D2" target="_self">Def2</a></i>;<br/><b>thus </b><a NAME="E5:105_1_1"/>
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font>
 <b>by </b><i>, , <a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2"><b>suppose </b></a><a NAME="E1:105_1_2"/><i><font color="Green">E94</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:105_1_2"/>
for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">v</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">G</font> being   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font>, <font color="Maroon">a</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E4:105_1_2"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">G</font>,<font color="Maroon">a</font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Maroon">g</font> &amp; <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">a</font> holds <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  ) ) )
 <b>by </b><i>, <a class="ref" href="mesfunc3.html#T14">MESFUNC3:14</a></i>;<br/><a NAME="E5:105_1_2"/><i><font color="Green">E98</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:105_1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:105_1_2_1"/><b>then </b><i><font color="Green">E99</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:105_1_2_1"/><i><font color="Green">E100</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:105_1_2_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:105_1_2_1_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:105_1_2_1_1"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T26" target="_self">Th26</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:105_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, <a class="txt" href="mesfunc5.html#E9:105_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:105_1_2_1"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E8:105_1_2"><i><font color="Green">E107</font></i></a>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a>, <a class="txt" href="mesfunc5.html#E9:105_1_2"><i><font color="Green">E108</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:105_1_2_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E7:105_1_2_1"/>
 ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T26" target="_self">Th26</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:105_1_2_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><a NAME="E6:105_1_2"/><i><font color="Green">E103</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> holds <br/> ex <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:105_1_2_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span>
 ;<br/>

<a NAME="E2:105_1_2_2"/><b>then </b><i><font color="Green">E104</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1_2_2_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:105_1_2_2_1_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>take </b><font color="Maroon">x</font> = 0;<br/><b>thus </b><a NAME="E2:105_1_2_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T26" target="_self">Th26</a>, <a class="txt" href="mesfunc5.html#E9:105_1_2"><i><font color="Green">E108</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:105_1_2_2_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:105_1_2_2_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:105_1_2_2_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:105_1_2_2_1_2"/><b>then </b>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E9:105_1_2"><i><font color="Green">E108</font></i></a>, </i>;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  <b>by </b><i><a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><b>take </b>
<font color="Maroon">x</font>
;<br/><b>thus </b><a NAME="E6:105_1_2_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon">a1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E8:105_1_2"/><i><font color="Green">E107</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>(<a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>);<br/></i><a NAME="E9:105_1_2"/><i><font color="Green">E108</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:105_1_2_3"/>
not 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:105_1_2_3"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_3"><i><font color="Green">E109</font></i></a>, <a class="ref" href="nat_1.html#T71">NAT_1:71</a></i>;<br/>
<a NAME="E3:105_1_2_3"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/>
<a NAME="E4:105_1_2_3"/><b>then </b><i><font color="Green">E109</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E9:105_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/>
<a NAME="E5:105_1_2_3"/><b>then </b><i><font color="Green">E121</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green">E122</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> = 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E9:105_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>
<br/>.= 
 <a href="tarski.html#K3">union</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T14">FUNCT_1:14</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>
;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:105_1_2_3"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:105_1_2"><i><font color="Green">E107</font></i></a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E9:105_1_2_3"/>
<font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E9:105_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, <a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:105_1_2_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, <a class="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma115</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:105_1_2"/><b>then </b><i><font color="Green">E124</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">a1</font>
 <b>by </b><i/>;<br/><a NAME="E11:105_1_2"/><b>then </b>
( <font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> 2 &amp; 2 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E12:105_1_2"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> 1
 <b>by </b><i><a class="txt" href="mesfunc5.html#E9:105_1_2"><i><font color="Green">E108</font></i></a></i>;<br/><a NAME="E13:105_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T26" target="_self">Th26</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E14:105_1_2"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span>
 ;<br/><a NAME="E15:105_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i/>;<br/><a NAME="E16:105_1_2"/><b>then </b>
( not <font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">a1</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">RINGA</font> = <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">a1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="membered.html#V2">real-membered</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>set </b><font color="Maroon">alpha</font> =  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span>;<br/><b>set </b><font color="Maroon">beta</font> =  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">beta1</font> =  <a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E19:105_1_2"/>
 <a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RINGA</font>
 <b>by </b><i><a class="ref" href="pre_circ.html#D4">PRE_CIRC:def 4</a></i>;<br/><a NAME="E20:105_1_2"/><b>then </b><i><font color="Green">E126</font></i>: 
(  <a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">a1</font> &amp; not  <a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E22:105_1_2"/><i><font color="Green">E127</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">a1</font> &amp;  <a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E24:105_1_2"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma115</font></i></a>, <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E25:105_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a1</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E26:105_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E27:105_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E28:105_1_2"/><b>then </b><i><font color="Green">E129</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E29:105_1_2"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">g</font> <a href="mesfunc2.html#R1">is_finite</a> 
 <b>by </b><i>, <a class="ref" href="mesfunc2.html#D5">MESFUNC2:def 5</a></i>;<br/><a NAME="E30:105_1_2"/><i><font color="Green">E131</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds   <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:105_1_2_4"/><i><font color="Green">E132</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:105_1_2_4"/>
( <font color="Maroon">g</font> is <a href="mesfunc5.html#V4" target="_self">without+infty</a> &amp; <font color="Maroon">g</font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:105_1_2_4"/><b>then </b>
( not  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> &amp; not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:105_1_2_4"/><b>then </b><i><font color="Green">E133</font></i>: 
( <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="funct_1.html#T142">FUNCT_1:142</a></i>;<br/>
<a NAME="E5:105_1_2_4"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:105_1_2_4"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E31:105_1_2"/><i><font color="Green">E134</font></i>: 
for <font color="Olive">e</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st  <a href="measure6.html#K1">R_EAL</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> &amp; <font color="Olive">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span> holds <br/> ex <font color="Olive">H</font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font> ex <font color="Olive">MF</font> being  <a href="mesfunc5.html#NM1">ExtREAL_sequence</a> st <br/>( ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Olive">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span></span>)</span>,<font color="Olive">e</font> ) &amp; ( for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Olive">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a> <font color="Olive">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> ) &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Olive">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> ) &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Olive">MF</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Olive">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> ) &amp; <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Olive">MF</font></span>)</span> &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Olive">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:105_1_2_5"/><i><font color="Green">E135</font></i>: 
(  <a href="measure6.html#K1">R_EAL</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> &amp; <font color="Maroon">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span> )
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> =  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">e</font>;<br/>
<b>consider </b><font color="Maroon">H</font> being   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:105_1_2_5"/><i><font color="Green">E136</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">H</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="kurato_2.html#S1">KURATO_2:sch 1</a>();<br/></i>
<div><i><font color="Green">E137</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:105_1_2_5_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E2:105_1_2_5_1"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font>)
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E5:105_1_2_5"/><i><font color="Green">E138</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:105_1_2_5_2"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:105_1_2_5_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:105_1_2_5_2_1"/><b>then </b><i><font color="Green">E140</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2"><i><font color="Green">E124</font></i></a></i>;<br/><a NAME="E3:105_1_2_5_2_1"/><b>then </b><i><font color="Green">E141</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp;  ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> ) )
 <b>by </b><i><a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/><a NAME="E4:105_1_2_5_2_1"/><b>then </b><i><font color="Green">E142</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>;<br/><a NAME="E5:105_1_2_5_2_1"/><i><font color="Green">E143</font></i>: 
 ex <font color="Olive">y1</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/><a NAME="E6:105_1_2_5_2_1"/><i><font color="Green">E184</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a></i>;<br/><a NAME="E7:105_1_2_5_2_1"/><b>then </b><i><font color="Green">E185</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E24:105_1_2"><i><font color="Green">E128</font></i></a>, <a class="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>;<br/><a NAME="E8:105_1_2_5_2_1"/><i><font color="Green">E186</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E20:105_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="mesfunc5.html#E24:105_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green">E131</font></i></a>, </i>;<br/><a NAME="E9:105_1_2_5_2_1"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green">E131</font></i></a>, <a class="txt" href="mesfunc5.html#E24:105_1_2"><i><font color="Green">E128</font></i></a>, <a class="ref" href="mesfunc5.html#T30" target="_self">Th30</a>, <a class="ref" href="mesfunc2.html#D1">MESFUNC2:def 1</a></i>;<br/><a NAME="E10:105_1_2_5_2_1"/><b>then </b>
( <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="ref" href="extreal2.html#T67">EXTREAL2:67</a></i>;<br/><a NAME="E11:105_1_2_5_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E28:105_1_2"><i><font color="Green">E129</font></i></a>, , , <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/><a NAME="E12:105_1_2_5_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/><a NAME="E13:105_1_2_5_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>,<font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E24:105_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green">E131</font></i></a>, <a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/><b>hence </b><a NAME="E14:105_1_2_5_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2"><i><font color="Green">E124</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:105_1_2_5_2"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:105_1_2_5_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">H</font>
 ;<br/><a NAME="E2:105_1_2_5_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K2">Union</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1">inferior_setsequence</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i><a class="ref" href="setlim_1.html#D4">SETLIM_1:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">V</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:105_1_2_5_3"/><i><font color="Green">E187</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1">inferior_setsequence</a> <font color="Maroon">H</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:105_1_2_5_3"/><i><font color="Green">E188</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1">inferior_setsequence</a> <font color="Maroon">H</font></span>)</span> &amp; <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1">inferior_setsequence</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E20:105_1_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a></i>;<br/><a NAME="E8:105_1_2_5_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0</span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E20:105_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="setlim_1.html#T24">SETLIM_1:24</a></i>;<br/><a NAME="E9:105_1_2_5_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2"><i><font color="Green">E124</font></i></a></i>;<br/><a NAME="E10:105_1_2_5_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/><b>hence </b><a NAME="E11:105_1_2_5_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:105_1_2_5"/><b>then </b><i><font color="Green">E189</font></i>: 
 <a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:105_1_2_5"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:105_1_2_5_4_1"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> ;<br/><a NAME="E3:105_1_2_5_4_1"/><i><font color="Green">E191</font></i>: 
( <font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp; <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a>, </i>;<br/><div><i><font color="Green">E192</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_1_2_5_4_1_1"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font> is <a href="mesfunc5.html#V9" target="_self">convergent_to_-infty</a>
 ;<br/><b>reconsider </b><font color="Maroon">E</font> = <font color="Maroon">e</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="measure5.html#T16">MEASURE5:16</a></i>;<br/><a NAME="E3:105_1_2_5_4_1_1"/><i><font color="Green">E194</font></i>: 
 <a href="real_1.html#K1">-</a> <font color="Maroon">E</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><b>then consider </b><font color="Maroon">N</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:105_1_2_5_4_1_1"/><i><font color="Green">E195</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">N</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">E</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green">E130</font></i></a>, </i>;<br/><a NAME="E6:105_1_2_5_4_1_1"/>
<font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">N</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E6:105_1_2"><i><font color="Green">E103</font></i></a></i>;<br/><a NAME="E7:105_1_2_5_4_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">N</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">N</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green">E131</font></i></a>, , <a class="ref" href="supinf_2.html#T70">SUPINF_2:70</a></i>;<br/><b>hence </b><a NAME="E8:105_1_2_5_4_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_5_4_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1_2_5_4_1_2_1"/>
( <font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font> is <a href="mesfunc5.html#V7" target="_self">convergent_to_finite_number</a> or <font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font> is <a href="mesfunc5.html#V8" target="_self">convergent_to_+infty</a> )
 </span><b>by </b><i><a class="txt" href="mesfunc5.html#E28:105_1_2"><i><font color="Green">E129</font></i></a>, <a class="txt" href="mesfunc5.html#E24:105_1_2"><i><font color="Green">E128</font></i></a>, <a class="ref" href="mesfunc5.html#D2" target="_self">Def2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_1"><b>suppose </b></a><a NAME="E1:105_1_2_5_4_1_2_1_1"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font> is <a href="mesfunc5.html#V7" target="_self">convergent_to_finite_number</a>
 ;<br/><div class="add"><a NAME="E2:105_1_2_5_4_1_2_1_1"/>
(  ex <font color="Olive">limFx</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>(  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">limFx</font> &amp; ( for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> holds <br/> ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">p</font> ) &amp; <font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font> is <a href="mesfunc5.html#V7" target="_self">convergent_to_finite_number</a> ) or (  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font> is <a href="mesfunc5.html#V8" target="_self">convergent_to_+infty</a> ) or (  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font> is <a href="mesfunc5.html#V9" target="_self">convergent_to_-infty</a> ) )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E24:105_1_2"><i><font color="Green">E128</font></i></a>, <a class="ref" href="mesfunc5.html#T14" target="_self">Th14</a></i>;<br/><b>then consider </b><font color="Maroon">limFx</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:105_1_2_5_4_1_2_1_1"/><i><font color="Green">E197</font></i>: 
(  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">limFx</font> &amp; ( for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> holds <br/> ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">p</font> ) )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="mesfunc5.html#D1" target="_self">Def1</a>, </i>;<br/><a NAME="E5:105_1_2_5_4_1_2_1_1"/>
 ex <font color="Olive">N</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">N</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">m</font></span>)</span></span>)</span>,<font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">E</font> = <font color="Maroon">e</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="measure5.html#T16">MEASURE5:16</a></i>;<br/>
<a NAME="E2:105_1_2_5_4_1_2_1_1_1"/><i><font color="Green">E198</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">E</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T217">XREAL_1:217</a></i>;<br/>
<b>set </b><font color="Maroon">E2</font> = <font color="Maroon">E</font> <a href="real_1.html#K6">/</a> 2;<br/>
<a NAME="E3:105_1_2_5_4_1_2_1_1_1"/>
<font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a></i>;<br/>
<a NAME="E4:105_1_2_5_4_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/>
<a NAME="E5:105_1_2_5_4_1_2_1_1_1"/><b>then </b><i><font color="Green">E199</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a>, <a class="ref" href="extreal1.html#T24">EXTREAL1:24</a></i>;<br/>
<b>consider </b><font color="Maroon">N</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:105_1_2_5_4_1_2_1_1_1"/><i><font color="Green">E200</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">N</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green">E131</font></i></a></i>;<br/>
<a NAME="E8:105_1_2_5_4_1_2_1_1_1"/><i><font color="Green">E201</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:105_1_2_5_4_1_2_1_1_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 ;<br/><b>then consider </b><font color="Maroon">N0</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:105_1_2_5_4_1_2_1_1_1_1_1"/><i><font color="Green">E202</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green">E131</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">N0</font> = <font color="Maroon">N0</font>, <font color="Maroon">n1</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>set </b><font color="Maroon">m</font> =  <a href="limfunc1.html#K1">max</a> <font color="Maroon">N0</font>,<font color="Maroon">n1</font>;<br/><a NAME="E5:105_1_2_5_4_1_2_1_1_1_1_1"/>
<font color="Maroon">N0</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">N0</font>,<font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="square_1.html#T46">SQUARE_1:46</a></i>;<br/><a NAME="E6:105_1_2_5_4_1_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E203</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">N0</font>,<font color="Maroon">n1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_5"><i><font color="Green">E136</font></i></a></i>;<br/><a NAME="E7:105_1_2_5_4_1_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">N0</font>,<font color="Maroon">n1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">N0</font>,<font color="Maroon">n1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span>
 <b>by </b><i><a class="ref" href="extreal2.html#T57">EXTREAL2:57</a></i>;<br/><a NAME="E8:105_1_2_5_4_1_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">N0</font>,<font color="Maroon">n1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_5"><i><font color="Green">E137</font></i></a></i>;<br/><a NAME="E9:105_1_2_5_4_1_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E204</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">N0</font>,<font color="Maroon">n1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">e</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="extreal2.html#T29">EXTREAL2:29</a>, <a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green">E131</font></i></a></i>;<br/><a NAME="E10:105_1_2_5_4_1_2_1_1_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">N0</font>,<font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="square_1.html#T46">SQUARE_1:46</a></i>;<br/><a NAME="E11:105_1_2_5_4_1_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">N0</font>,<font color="Maroon">n1</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a></i>;<br/><a NAME="E12:105_1_2_5_4_1_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">N0</font>,<font color="Maroon">n1</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E13:105_1_2_5_4_1_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">N0</font>,<font color="Maroon">n1</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>;<br/><b>hence </b><a NAME="E14:105_1_2_5_4_1_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">e</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:105_1_2_5"><i><font color="Green">E138</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:105_1_2_5_4_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E205</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:105_1_2_5_4_1_2_1_1_1_1"/>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a>, <a class="ref" href="supinf_1.html#T21">SUPINF_1:21</a></i>;<br/>
<a NAME="E4:105_1_2_5_4_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E206</font></i>: 
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a>, <a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green">E131</font></i></a></i>;<br/>
<a NAME="E5:105_1_2_5_4_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_5"><i><font color="Green">E136</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E6:105_1_2_5_4_1_2_1_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_5"><i><font color="Green">E136</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_5"><i><font color="Green">E137</font></i></a>, <a class="ref" href="extreal1.html#T40">EXTREAL1:40</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E207</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:105_1_2_5_4_1_2_1_1_1_2"/><i><font color="Green">E208</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5"><i><font color="Green">E135</font></i></a></i>;<br/><b>thus </b><span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> = 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_5"><i><font color="Green">E137</font></i></a>, <a class="ref" href="extreal1.html#D3">EXTREAL1:def 3</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E209</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:105_1_2_5_4_1_2_1_1_1_3"/><i><font color="Green">E210</font></i>: 
<font color="Maroon">N</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:105_1_2_5_4_1_2_1_1_1_3"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E31:105_1_2"><i><font color="Green">E134</font></i></a></i>;<br/><a NAME="E3:105_1_2_5_4_1_2_1_1_1_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green">E131</font></i></a>, </i>;<br/><a NAME="E4:105_1_2_5_4_1_2_1_1_1_3"/><b>then </b><i><font color="Green">E211</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E5:105_1_2_5_4_1_2_1_1_1_3"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_5"><i><font color="Green">E136</font></i></a></i>;<br/><a NAME="E6:105_1_2_5_4_1_2_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:105_1_2_5"><i><font color="Green">E138</font></i></a>, <a class="txt" href="mesfunc5.html#E31:105_1_2"><i><font color="Green">E134</font></i></a></i>;<br/><a NAME="E7:105_1_2_5_4_1_2_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>;<br/><b>hence </b><a NAME="E8:105_1_2_5_4_1_2_1_1_1_3"/>
 <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="extreal2.html#T29">EXTREAL2:29</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:105_1_2_5_4_1_2_1_1_1_4"/><i><font color="Green">E305</font></i>: 
<font color="Maroon">N</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:105_1_2_5_4_1_2_1_1_1_4"/><i><font color="Green">E306</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a>, <a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a></i>;<br/><a NAME="E3:105_1_2_5_4_1_2_1_1_1_4"/><i><font color="Green">E307</font></i>: 
<font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="extreal2.html#T33">EXTREAL2:33</a></i>;<br/><a NAME="E4:105_1_2_5_4_1_2_1_1_1_4"/>
 <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:105_1_2_5"><i><font color="Green">E137</font></i></a>, <a class="txt" href="mesfunc5.html#E5:105_1_2_5"><i><font color="Green">E138</font></i></a></i>;<br/><a NAME="E5:105_1_2_5_4_1_2_1_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon">E</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:105_1_2_5_4_1_2_1_1_1_4"/><b>then </b>
<font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon">E</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:105_1_2_5_4_1_2_1_1_1_4"/><b>then </b>
<font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon">E</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon">E</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T8">EXTREAL1:8</a></i>;<br/><a NAME="E8:105_1_2_5_4_1_2_1_1_1_4"/><b>then </b>
<font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">E</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">E</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E9:105_1_2_5_4_1_2_1_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="extreal2.html#T30">EXTREAL2:30</a></i>;<br/><a NAME="E10:105_1_2_5_4_1_2_1_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>;<br/><b>hence </b><a NAME="E11:105_1_2_5_4_1_2_1_1_1_4"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>,<font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:105_1_2_5_4_1_2_1_1_1"/>
 ex <font color="Olive">N</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">N</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">m</font></span>)</span></span>)</span>,<font color="Maroon">e</font>
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">N</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:105_1_2_5_4_1_2_1_1"/><i><font color="Green">E308</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">N</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">m</font></span>)</span></span>)</span>,<font color="Maroon">e</font>
 ;<br/><b>reconsider </b><font color="Maroon">N</font> = <font color="Maroon">N</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:105_1_2_5_4_1_2_1_1_2"/>
<font color="Maroon">N</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E2:105_1_2_5_4_1_2_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">N</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">e</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:105_1_2_5_4_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2"><i><font color="Green">E124</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E10:105_1_2_5_4_1_2_1_1"/><b>then </b><i><font color="Green">E309</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1">inferior_setsequence</a> <font color="Maroon">H</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="setlim_1.html#T24">SETLIM_1:24</a></i>;<br/><a NAME="E11:105_1_2_5_4_1_2_1_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1">inferior_setsequence</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E12:105_1_2_5_4_1_2_1_1"/>
 ex <font color="Olive">N</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">N</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1">inferior_setsequence</a> <font color="Maroon">H</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1">inferior_setsequence</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">N</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_2"><b>suppose </b></a><a NAME="E1:105_1_2_5_4_1_2_1_2"/><i><font color="Green">E310</font></i>: 
<font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font> is <a href="mesfunc5.html#V8" target="_self">convergent_to_+infty</a>
 ;<br/><div class="add"><a NAME="E2:105_1_2_5_4_1_2_1_2"/>
 ex <font color="Olive">N</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">N</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:105_1_2_5_4_1_2_1_2_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 
<b>by </b><i>, <a class="ref" href="measure5.html#T16">MEASURE5:16</a></i>;<br/>
<a NAME="E2:105_1_2_5_4_1_2_1_2_1"/><b>then </b><i><font color="Green">E311</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 
<b>by </b><i><a class="ref" href="supinf_1.html#T17">SUPINF_1:17</a>, <a class="ref" href="supinf_1.html#T18">SUPINF_1:18</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1_2_5_4_1_2_1_2_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">e</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:105_1_2_5_4_1_2_1_2_1_1_1"/><i><font color="Green">E312</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">N</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:105_1_2_5_4_1_2_1_2_1_1_1"/><i><font color="Green">E313</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">N</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="measure6.html#K1">R_EAL</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="mesfunc5.html#T13" target="_self">Th13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:105_1_2_5_4_1_2_1_2_1_1_1_1"/>
<font color="Maroon">N</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:105_1_2_5_4_1_2_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:105_1_2_5_4_1_2_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green">E131</font></i></a>, <a class="ref" href="extreal2.html#T29">EXTREAL2:29</a></i>;<br/><a NAME="E4:105_1_2_5_4_1_2_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green">E131</font></i></a>, <a class="ref" href="extreal2.html#T30">EXTREAL2:30</a></i>;<br/><b>hence </b><a NAME="E5:105_1_2_5_4_1_2_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:105_1_2_5_4_1_2_1_2_1_1_1"/>
 ex <font color="Olive">N</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">N</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:105_1_2_5_4_1_2_1_2_1_1_2"/><i><font color="Green">E314</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">e</font>
 ;<br/><div class="add"><a NAME="E2:105_1_2_5_4_1_2_1_2_1_1_2"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="mesfunc5.html#T30" target="_self">Th30</a>, <a class="ref" href="mesfunc2.html#D1">MESFUNC2:def 1</a></i>;<br/><a NAME="E3:105_1_2_5_4_1_2_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="ref" href="extreal2.html#T67">EXTREAL2:67</a></i>;<br/><b>then reconsider </b><font color="Maroon">gx1</font> = <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><b>reconsider </b><font color="Maroon">e1</font> = <font color="Maroon">e</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="measure5.html#T16">MEASURE5:16</a></i>;<br/><a NAME="E6:105_1_2_5_4_1_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">gx1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">e1</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/><b>then reconsider </b><font color="Maroon">ee</font> = <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">e</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E8:105_1_2_5_4_1_2_1_2_1_1_2"/><i><font color="Green">E315</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ee</font> <a href="real_1.html#K3">+</a> 1 &amp; <font color="Maroon">ee</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ee</font> <a href="real_1.html#K3">+</a> 1 )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E9:105_1_2_5_4_1_2_1_2_1_1_2"/><i><font color="Green">E316</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">ee</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">ee</font> <a href="real_1.html#K3">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><b>consider </b><font color="Maroon">N</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:105_1_2_5_4_1_2_1_2_1_1_2"/><i><font color="Green">E317</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">N</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">ee</font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="mesfunc5.html#T13" target="_self">Th13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:105_1_2_5_4_1_2_1_2_1_1_2_1"/>
<font color="Maroon">N</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:105_1_2_5_4_1_2_1_2_1_1_2_1"/><b>then </b>
 <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">ee</font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5"><i><font color="Green">E135</font></i></a></i>;<br/><a NAME="E3:105_1_2_5_4_1_2_1_2_1_1_2_1"/><b>then </b>
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">ee</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E31:105_1_2"><i><font color="Green">E134</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:105_1_2_5_4_1_2_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green">E131</font></i></a>, <a class="ref" href="extreal2.html#T29">EXTREAL2:29</a></i>;<br/><a NAME="E5:105_1_2_5_4_1_2_1_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green">E131</font></i></a>, <a class="ref" href="extreal2.html#T30">EXTREAL2:30</a></i>;<br/><b>hence </b><a NAME="E6:105_1_2_5_4_1_2_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:105_1_2_5_4_1_2_1_2_1_1_2"/>
 ex <font color="Olive">N</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">N</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then consider </b><font color="Maroon">N</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:105_1_2_5_4_1_2_1_2"/><i><font color="Green">E318</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">N</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 ;<br/><b>reconsider </b><font color="Maroon">N</font> = <font color="Maroon">N</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><div><i><font color="Green">E319</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:105_1_2_5_4_1_2_1_2_2"/>
<font color="Maroon">N</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:105_1_2_5_4_1_2_1_2_2"/><b>then </b><i><font color="Green">E320</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green">E131</font></i></a></i>;<br/><a NAME="E3:105_1_2_5_4_1_2_1_2_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a>, <a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a></i>;<br/><a NAME="E4:105_1_2_5_4_1_2_1_2_2"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> )
 <b>by </b><i>, <a class="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>;<br/><b>hence </b><a NAME="E5:105_1_2_5_4_1_2_1_2_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>,<font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_4_1_2_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:105_1_2_5_4_1_2_1_2_3"/>
<font color="Maroon">N</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E2:105_1_2_5_4_1_2_1_2_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">N</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">e</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:105_1_2_5_4_1_2_1_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2"><i><font color="Green">E124</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E8:105_1_2_5_4_1_2_1_2"/><b>then </b><i><font color="Green">E321</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1">inferior_setsequence</a> <font color="Maroon">H</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="setlim_1.html#T24">SETLIM_1:24</a></i>;<br/><a NAME="E9:105_1_2_5_4_1_2_1_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1">inferior_setsequence</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E10:105_1_2_5_4_1_2_1_2"/>
 ex <font color="Olive">N</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">N</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1">inferior_setsequence</a> <font color="Maroon">H</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1">inferior_setsequence</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">N</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon">N</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:105_1_2_5_4_1"/><i><font color="Green">E322</font></i>: 
( <font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1">inferior_setsequence</a> <font color="Maroon">H</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1">inferior_setsequence</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">N</font> )
 ;<br/><a NAME="E8:105_1_2_5_4_1"/>
<span class="p1">(<span class="default"><a href="setlim_1.html#K1">inferior_setsequence</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1">inferior_setsequence</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E9:105_1_2_5_4_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K2">Union</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1">inferior_setsequence</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E10:105_1_2_5_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="setlim_1.html#D4">SETLIM_1:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:105_1_2_5_4"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:105_1_2_5"/><b>then </b><i><font color="Green">E323</font></i>: 
 <a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E20:105_1_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E10:105_1_2_5"/><i><font color="Green">E324</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:105_1_2_5_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:105_1_2_5_5_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2"><i><font color="Green">E124</font></i></a></i>;<br/><a NAME="E3:105_1_2_5_5_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/><b>hence </b><a NAME="E4:105_1_2_5_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:105_1_2_5_5"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:105_1_2_5"/>
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">H</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> <a href="prob_1.html#K1">.</a> <font color="Olive">m</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E12:105_1_2"><i><font color="Green">E125</font></i></a></i>;<br/>
<a NAME="E12:105_1_2_5"/><b>then </b><i><font color="Green">E325</font></i>: 
<font color="Maroon">H</font> is <a href="prob_1.html#V3">non-decreasing</a>
 
<b>by </b><i><a class="ref" href="prob_1.html#D7">PROB_1:def 7</a></i>;<br/>
<a NAME="E13:105_1_2_5"/><i><font color="Green">E326</font></i>: 
 <a href="prob_1.html#K2">Union</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:105_1_2_5_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K2">Union</a> <font color="Maroon">H</font>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:105_1_2_5_6"/><i><font color="Green">E327</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/>
<a NAME="E4:105_1_2_5_6"/>
<font color="Maroon">H</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E24:105_1_2"><i><font color="Green">E128</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:105_1_2_5_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green">E131</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:105_1_2_5"/><i><font color="Green">E328</font></i>: 
 <a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a>  <a href="kurato_2.html#K5">lim_sup</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="kurato_2.html#T9">KURATO_2:9</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  = <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">MF</font> being   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E16:105_1_2_5"/><i><font color="Green">E329</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">MF</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<div><i><font color="Green">E330</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:105_1_2_5_7"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E2:105_1_2_5_7"/>
<font color="Maroon">MF</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">n</font>)
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E18:105_1_2_5"/><i><font color="Green">E331</font></i>: 
( <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">MF</font></span>)</span> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">H</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_8"><b>proof </b></a><div class="add">

<div><i><font color="Green">E332</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:105_1_2_5_8_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:105_1_2_5_8_1"/><i><font color="Green">E334</font></i>: 
<font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">GF</font> being   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E5:105_1_2_5_8_1"/><i><font color="Green">E336</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">GF</font></span>)</span> &amp; ( for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a><br/> for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">GF</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GF</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GF</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">y</font> ) )
 <b>by </b><i><a class="ref" href="mesfunc2.html#D5">MESFUNC2:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">DGH</font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">GF</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="mesfunc2.html#T34">MESFUNC2:34</a></i>;<br/><a NAME="E7:105_1_2_5_8_1"/><i><font color="Green">E337</font></i>: 
( <font color="Maroon">g</font> <a href="mesfunc2.html#R1">is_finite</a>  &amp; <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R1">is_finite</a>  )
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E3:105_1_2_5"><i><font color="Green">E136</font></i></a>, <a class="ref" href="mesfunc2.html#D5">MESFUNC2:def 5</a></i>;<br/><a NAME="E8:105_1_2_5_8_1"/><i><font color="Green">E338</font></i>: 
<font color="Maroon">g</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">DGH</font>
 <b>by </b><i>, <a class="ref" href="mesfunc2.html#T37">MESFUNC2:37</a></i>;<br/><b>reconsider </b><font color="Maroon">E</font> = <font color="Maroon">e</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="measure5.html#T16">MEASURE5:16</a></i>;<br/><a NAME="E10:105_1_2_5_8_1"/>
<font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">DGH</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:105_1_2_5"><i><font color="Green">E136</font></i></a>, <a class="ref" href="mesfunc2.html#T37">MESFUNC2:37</a></i>;<br/><a NAME="E11:105_1_2_5_8_1"/><b>then </b>
<font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">DGH</font>
 <b>by </b><i><a class="ref" href="mesfunc2.html#T13">MESFUNC2:13</a>, , <a class="txt" href="mesfunc5.html#E5:105_1_2_5"><i><font color="Green">E138</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_5"><i><font color="Green">E137</font></i></a></i>;<br/><a NAME="E12:105_1_2_5_8_1"/><b>then </b>
<font color="Maroon">DGH</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">E</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1">is_measurable_on</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D17">MESFUNC1:def 17</a></i>;<br/><a NAME="E13:105_1_2_5_8_1"/><b>then </b><i><font color="Green">E339</font></i>: 
<font color="Maroon">DGH</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">E</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D11">MESFUNC1:def 11</a></i>;<br/><a NAME="E14:105_1_2_5_8_1"/><i><font color="Green">E340</font></i>: 
 <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">E</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_8_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:105_1_2_5_8_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">E</font></span>)</span>
 ;<br/>

<b>hence </b><a NAME="E2:105_1_2_5_8_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:105_1_2_5_8_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/><a NAME="E16:105_1_2_5_8_1"/><b>then </b>
<font color="Maroon">DGH</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">E</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">E</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_5"><i><font color="Green">E137</font></i></a></i>;<br/><a NAME="E17:105_1_2_5_8_1"/><b>then </b>
<font color="Maroon">DGH</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">E</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">E</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a>, </i>;<br/><b>hence </b><a NAME="E18:105_1_2_5_8_1"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E10:105_1_2"><i><font color="Green">E124</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:105_1_2_5_8"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">HH</font> = <font color="Maroon">H</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">S</font> <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5"><i><font color="Green">E135</font></i></a>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<div><i><font color="Green">E341</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_8_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:105_1_2_5_8_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>hence </b><a NAME="E2:105_1_2_5_8_2"/>
<font color="Maroon">HH</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">HH</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E12:105_1_2"><i><font color="Green">E125</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:105_1_2_5_8"/>
 <a href="kurato_2.html#K5">lim_sup</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="prob_1.html#K2">Union</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="setlim_1.html#T65">SETLIM_1:65</a>, <a class="txt" href="mesfunc5.html#E28:105_1_2"><i><font color="Green">E129</font></i></a></i>;<br/>
<a NAME="E6:105_1_2_5_8"/><b>then </b><i><font color="Green">E342</font></i>: 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green">E131</font></i></a>, <a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a>, <a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E7:105_1_2_5_8"/><i><font color="Green">E343</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">MF</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:105_1_2_5_8"/>
 <a href="measure1.html#K5">rng</a> <font color="Maroon">HH</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E9:105_1_2_5_8"/><b>then </b><i><font color="Green">E344</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:105_1_2_5_8"/><i><font color="Green">E345</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">MF</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</font> &amp; <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">M</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_8_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_8_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_1_2_5_8_3_1"/><i><font color="Green">E346</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">MF</font>
 ;<br/><a NAME="E2:105_1_2_5_8_3_1"/><b>then </b>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:105_1_2_5"><i><font color="Green">E138</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E3:105_1_2_5_8_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</font> &amp; <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">M</font> )
 <b>by </b><i>, , <a class="txt" href="mesfunc5.html#E5:105_1_2_5"><i><font color="Green">E138</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:105_1_2_5_8_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">MF</font> iff ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</font> &amp; <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">M</font> ) )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:105_1_2_5"><i><font color="Green">E138</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:105_1_2_5_8"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">MF</font> holds <br/><font color="Maroon">MF</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:105_1_2_5_8"/><b>then </b>
<font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MF</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T20">FUNCT_1:20</a></i>;<br/>
<b>hence </b><a NAME="E13:105_1_2_5_8"/>
( <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">MF</font></span>)</span> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">H</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2_5"><i><font color="Green">E135</font></i></a>, <a class="txt" href="mesfunc5.html#E3:105_1_2_5"><i><font color="Green">E136</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2_5"><i><font color="Green">E137</font></i></a>, <a class="ref" href="measure2.html#T27">MEASURE2:27</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_5_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:105_1_2_5_9"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E2:105_1_2_5_9"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E31:105_1_2"><i><font color="Green">E134</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E20:105_1_2_5"/>
 ex <font color="Olive">H</font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font> ex <font color="Olive">MF</font> being  <a href="mesfunc5.html#NM1">ExtREAL_sequence</a> st <br/>( ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Olive">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span></span>)</span>,<font color="Maroon">e</font> ) &amp; ( for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Olive">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a> <font color="Olive">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> ) &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Olive">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> ) &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Olive">MF</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Olive">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> ) &amp; <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Olive">MF</font></span>)</span> &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Olive">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E24:105_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="mesfunc5.html#E10:105_1_2"><i><font color="Green">E124</font></i></a>, <a class="txt" href="mesfunc5.html#E12:105_1_2"><i><font color="Green">E125</font></i></a>, <a class="txt" href="mesfunc5.html#E31:105_1_2"><i><font color="Green">E134</font></i></a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E32:105_1_2"/><i><font color="Green">E347</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E33:105_1_2"/><b>then </b><i><font color="Green">E348</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E9:105_1_2"><i><font color="Green">E108</font></i></a>, , <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E34:105_1_2"/><i><font color="Green">E349</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/>(  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> &amp; <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:105_1_2_6"/><i><font color="Green">E350</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:105_1_2_6"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E9:105_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:105_1_2_6"/><i><font color="Green">E351</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:105_1_2_6"/><i><font color="Green">E352</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E20:105_1_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a></i>;<br/>
<a NAME="E8:105_1_2_6"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E9:105_1_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/>
<a NAME="E9:105_1_2_6"/><b>then </b><i><font color="Green">E353</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:105_1_2_6"/><b>then </b><i><font color="Green">E354</font></i>: 
<font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">a1</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_1_2_6_1"/><i><font color="Green">E356</font></i>: 
<font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> 1
 ;<br/><a NAME="E2:105_1_2_6_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T26" target="_self">Th26</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:105_1_2_6_1"/><b>then </b><i><font color="Green">E357</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">a1</font>
 <b>by </b><i/>;<br/><a NAME="E4:105_1_2_6_1"/>
<font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E24:105_1_2"><i><font color="Green">E128</font></i></a></i>;<br/><a NAME="E5:105_1_2_6_1"/><b>then </b><i><font color="Green">E358</font></i>: 
<font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> 1
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green">E131</font></i></a>, , <a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green">E130</font></i></a></i>;<br/><a NAME="E6:105_1_2_6_1"/>
<font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E20:105_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a>, <a class="txt" href="mesfunc5.html#E9:105_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><b>hence </b><a NAME="E7:105_1_2_6_1"/>
contradiction
 <b>by </b><i><a class="txt" href="mesfunc5.html#E12:105_1_2"><i><font color="Green">E125</font></i></a>, , , <a class="txt" href="mesfunc5.html#E9:105_1_2"><i><font color="Green">E108</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:105_1_2_6"/><b>then </b>
not <font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E13:105_1_2_6"/><b>then </b><i><font color="Green">E359</font></i>: 
<font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RINGA</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E28:105_1_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E20:105_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a>, <a class="txt" href="mesfunc5.html#E9:105_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>
<br/>.= 
<font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E24:105_1_2"><i><font color="Green">E128</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E15:105_1_2_6"/>
(  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a>, <a class="ref" href="pre_circ.html#D4">PRE_CIRC:def 4</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1_2_7"/>
( <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7_1"><b>suppose </b></a><a NAME="E1:105_1_2_7_1"/><i><font color="Green">E360</font></i>: 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">DG</font> =  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i>, </i>;<br/><a NAME="E3:105_1_2_7_1"/><i><font color="Green">E361</font></i>: 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="measure1.html#D11">MEASURE1:def 11</a></i>;<br/><a NAME="E4:105_1_2_7_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="measure1.html#R2">is_measurable</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#T66">MEASURE1:66</a></i>;<br/><a NAME="E5:105_1_2_7_1"/><b>then </b><i><font color="Green">E362</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="measure1.html#D12">MEASURE1:def 12</a></i>;<br/><a NAME="E6:105_1_2_7_1"/>
(  <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> )
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><a NAME="E7:105_1_2_7_1"/><b>then </b><i><font color="Green">E363</font></i>: 
( <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E12:105_1_2"><i><font color="Green">E125</font></i></a>, <a class="txt" href="mesfunc5.html#E20:105_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/><a NAME="E8:105_1_2_7_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 <b>by </b><i/>;<br/><a NAME="E9:105_1_2_7_1"/><b>then </b><i><font color="Green">E364</font></i>: 
<font color="Maroon">g</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#T71">SUPINF_2:71</a></i>;<br/><a NAME="E10:105_1_2_7_1"/><i><font color="Green">E365</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:105_1_2_7_1_1"/><i><font color="Green">E366</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:105_1_2"><i><font color="Green">E107</font></i></a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E3:105_1_2_7_1_1"/>
(  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2"><i><font color="Green">E124</font></i></a>, <a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green">E131</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:105_1_2_7_1_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E9:105_1_2"><i><font color="Green">E108</font></i></a>, , <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:105_1_2_7_1"/><i><font color="Green">E367</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">DG</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">GP</font> being   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E2:105_1_2_7_1_2"/><i><font color="Green">E368</font></i>: 
( <font color="Maroon">GP</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">GP</font> <a href="hidden.html#R1">=</a> <font color="Maroon">DG</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DG</font> holds <br/><font color="Maroon">GP</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span> ) )
 <b>by </b><i/>;<br/>
<a NAME="E3:105_1_2_7_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">GP</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">GP</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green">E130</font></i></a></i>;<br/>
<a NAME="E4:105_1_2_7_1_2"/><b>then </b><i><font color="Green">E369</font></i>: 
<font color="Maroon">GP</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T71">SUPINF_2:71</a></i>;<br/>
<a NAME="E5:105_1_2_7_1_2"/><i><font color="Green">E370</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">GP</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span> holds <br/><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">GP</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">GP</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span>
 ;<br/>

<a NAME="E2:105_1_2_7_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">GP</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">GP</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">GP</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E3:105_1_2_7_1_2_1"/><b>then </b><i><font color="Green">E371</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">GP</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:105_1_2_7_1_2_1"/><b>then </b>
<font color="Maroon">GP</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:105_1_2_7_1_2_1"/>
<font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">GP</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2"><i><font color="Green">E124</font></i></a>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:105_1_2_7_1_2"/><b>then </b><i><font color="Green">E372</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">GP</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">GP</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">GP</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2"><i><font color="Green">E94</font></i></a>, , <a class="txt" href="mesfunc5.html#E28:105_1_2"><i><font color="Green">E129</font></i></a>, , </i>;<br/>
<a NAME="E7:105_1_2_7_1_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">GP</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">GP</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i>, , , <a class="txt" href="mesfunc5.html#E28:105_1_2"><i><font color="Green">E129</font></i></a>, <a class="txt" href="mesfunc5.html#E31:105_1_2"><i><font color="Green">E134</font></i></a>, </i>;<br/>
<a NAME="E8:105_1_2_7_1_2"/><b>then </b>
( <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">GP</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">GP</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">GP</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">GP</font> )
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T64">GRFUNC_1:64</a>, </i>;<br/>
<b>hence </b><a NAME="E9:105_1_2_7_1_2"/>
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">DG</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green">E130</font></i></a>, , <a class="txt" href="mesfunc5.html#E4:105_1_2"><i><font color="Green">E95</font></i></a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">MG</font> = <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="mesfunc5.html#E24:105_1_2"><i><font color="Green">E128</font></i></a>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><a NAME="E13:105_1_2_7_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">DG</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">beta1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">MG</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T13">EXTREAL1:13</a></i>;<br/><a NAME="E14:105_1_2_7_1"/><b>then </b><i><font color="Green">E373</font></i>: 
(  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T24" target="_self">Th24</a>, , <a class="txt" href="mesfunc5.html#E28:105_1_2"><i><font color="Green">E129</font></i></a>, <a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green">E131</font></i></a>, <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/><a NAME="E15:105_1_2_7_1"/><i><font color="Green">E375</font></i>: 
for <font color="Olive">e</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> &amp; <font color="Olive">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span> holds <br/> ex <font color="Olive">N0</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Olive">e</font> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_1_3"/><i><font color="Green">E376</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> &amp; <font color="Maroon">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span> )
 ;<br/>

<a NAME="E2:105_1_2_7_1_3"/><b>then </b><i><font color="Green">E377</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T4">XXREAL_0:4</a></i>;<br/>
<div><i><font color="Green">E378</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:105_1_2_7_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 ;<br/><a NAME="E2:105_1_2_7_1_3_1"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="mesfunc5.html#T30" target="_self">Th30</a>, <a class="ref" href="mesfunc2.html#D1">MESFUNC2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:105_1_2_7_1_3_1"/>
( <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="ref" href="extreal2.html#T67">EXTREAL2:67</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">H</font> being   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>, <font color="Maroon">MF</font> being   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E5:105_1_2_7_1_3"/><i><font color="Green">E379</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span></span>)</span>,<font color="Maroon">e</font>
 <b>and </b><br/><a NAME="E6:105_1_2_7_1_3"/><i><font color="Green">E380</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font>
 <b>and </b><br/><a NAME="E7:105_1_2_7_1_3"/><i><font color="Green">E381</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E8:105_1_2_7_1_3"/><i><font color="Green">E382</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">MF</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span>
 <b>and </b><br/><a NAME="E9:105_1_2_7_1_3"/><i><font color="Green">E383</font></i>: 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">MF</font></span>)</span>
 <b>and </b><br/><a NAME="E10:105_1_2_7_1_3"/><i><font color="Green">E384</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E31:105_1_2"><i><font color="Green">E134</font></i></a>, </i>;<br/>
<a NAME="E11:105_1_2_7_1_3"/>
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">MF</font></span>)</span> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E24:105_1_2"><i><font color="Green">E128</font></i></a>, , <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E13:105_1_2_7_1_3"/><i><font color="Green">E385</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">MF</font> &amp; <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">MF</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E31:105_1_2"><i><font color="Green">E134</font></i></a>, <a class="ref" href="measure6.html#T33">MEASURE6:33</a></i>;<br/>
<b>consider </b><font color="Maroon">N0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:105_1_2_7_1_3"/><i><font color="Green">E386</font></i>: 
( <font color="Maroon">N0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">MF</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MF</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">N0</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">N0</font> = <font color="Maroon">N0</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B0</font> = <font color="Maroon">H</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">N0</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E18:105_1_2_7_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">N0</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a>, </i>;<br/>
<a NAME="E19:105_1_2_7_1_3"/><b>then </b>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">N0</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">e</font>
 
<b>by </b><i>, <a class="ref" href="extreal2.html#T29">EXTREAL2:29</a></i>;<br/>
<a NAME="E20:105_1_2_7_1_3"/><b>then </b><i><font color="Green">E415</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">N0</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 
<b>by </b><i>, <a class="ref" href="extreal2.html#T30">EXTREAL2:30</a></i>;<br/>
<a NAME="E21:105_1_2_7_1_3"/><i><font color="Green">E416</font></i>: 
<font color="Maroon">B0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">DG</font>
 
<b>by </b><i/>;<br/>
<a NAME="E22:105_1_2_7_1_3"/><b>then </b><i><font color="Green">E417</font></i>: 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">DG</font>
 
<b>by </b><i><a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/>
<a NAME="E23:105_1_2_7_1_3"/>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">DG</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/>
<a NAME="E24:105_1_2_7_1_3"/><b>then </b>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">DG</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E12:105_1_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/>
<a NAME="E25:105_1_2_7_1_3"/><b>then </b>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E26:105_1_2_7_1_3"/><b>then </b><i><font color="Green">E418</font></i>: 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">DG</font> <a href="measure1.html#K4">\</a> <font color="Maroon">B0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">DG</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a>, <a class="ref" href="measure1.html#T63">MEASURE1:63</a></i>;<br/>
<b>take </b>
<font color="Maroon">N0</font>
;<br/>

<div><i><font color="Green">E419</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:105_1_2_7_1_3_2"/><i><font color="Green">E420</font></i>: 
<font color="Maroon">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><b>reconsider </b><font color="Maroon">BN</font> = <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/><a NAME="E3:105_1_2_7_1_3_2"/>
<font color="Maroon">H</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">N0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:105_1_2_7_1_3_2"/><b>then </b>
<font color="Maroon">DG</font> <a href="measure1.html#K4">\</a> <font color="Maroon">BN</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">DG</font> <a href="measure1.html#K4">\</a> <font color="Maroon">B0</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T34">XBOOLE_1:34</a></i>;<br/><a NAME="E5:105_1_2_7_1_3_2"/><b>then </b>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">DG</font> <a href="measure1.html#K4">\</a> <font color="Maroon">BN</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">DG</font> <a href="measure1.html#K4">\</a> <font color="Maroon">B0</font></span>)</span>
 <b>by </b><i><a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/><b>hence </b><a NAME="E6:105_1_2_7_1_3_2"/>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:105_1_2_7_1_3_3"/><i><font color="Green">E421</font></i>: 
<font color="Maroon">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:105_1_2_7_1_3_3"/><i><font color="Green">E422</font></i>: 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">DG</font>
 <b>by </b><i/>;<br/><a NAME="E3:105_1_2_7_1_3_3"/><b>then </b>
<font color="Maroon">DG</font> <a href="hidden.html#R1">=</a> <font color="Maroon">DG</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><a NAME="E4:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E423</font></i>: 
<font color="Maroon">DG</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">DG</font> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>;<br/><a NAME="E5:105_1_2_7_1_3_3"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font>
 <b>by </b><i/>;<br/><a NAME="E6:105_1_2_7_1_3_3"/><b>then </b>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="measure1.html#R2">is_measurable</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#D12">MEASURE1:def 12</a></i>;<br/><a NAME="E7:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E424</font></i>: 
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure1.html#R2">is_measurable</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#T66">MEASURE1:66</a></i>;<br/><a NAME="E8:105_1_2_7_1_3_3"/>
<font color="Maroon">DG</font> <a href="measure1.html#R2">is_measurable</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#D12">MEASURE1:def 12</a></i>;<br/><a NAME="E9:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E425</font></i>: 
<font color="Maroon">DG</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="measure1.html#R2">is_measurable</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#T66">MEASURE1:66</a>, </i>;<br/><a NAME="E10:105_1_2_7_1_3_3"/>
<font color="Maroon">DG</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">DG</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T49">XBOOLE_1:49</a></i>;<br/><a NAME="E11:105_1_2_7_1_3_3"/><b>then </b>
<font color="Maroon">DG</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">DG</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">DG</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="mesfunc5.html#T36" target="_self">Th36</a>, <a class="ref" href="measure1.html#D12">MEASURE1:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/><a NAME="E14:105_1_2_7_1_3_3"/><i><font color="Green">E426</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/><a NAME="E15:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E427</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E16:105_1_2_7_1_3_3"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 ;<br/><a NAME="E17:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E429</font></i>: 
<font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E18:105_1_2_7_1_3_3"/><i><font color="Green">E430</font></i>: 
<font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E6:105_1_2"><i><font color="Green">E103</font></i></a></i>;<br/><a NAME="E19:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E431</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 <b>by </b><i/>;<br/><a NAME="E20:105_1_2_7_1_3_3"/><i><font color="Green">E432</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><a NAME="E21:105_1_2_7_1_3_3"/><i><font color="Green">E433</font></i>: 
<font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/><a NAME="E22:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E434</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="txt" href="mesfunc5.html#E46"><i><font color="Green">Lemma157</font></i></a></i>;<br/><a NAME="E23:105_1_2_7_1_3_3"/><i><font color="Green">E435</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E24:105_1_2_7_1_3_3"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T24" target="_self">Th24</a>, </i>;<br/><a NAME="E25:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E436</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T39" target="_self">Th39</a>, <a class="ref" href="supinf_2.html#T20">SUPINF_2:20</a></i>;<br/><a NAME="E26:105_1_2_7_1_3_3"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E27:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E437</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T37" target="_self">Th37</a>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><b>consider </b><font color="Maroon">EP</font> being   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E29:105_1_2_7_1_3_3"/><i><font color="Green">E438</font></i>: 
( <font color="Maroon">EP</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">EP</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">EP</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> ) )
 <b>by </b><i>, </i>;<br/><a NAME="E30:105_1_2_7_1_3_3"/><i><font color="Green">E439</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">EP</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E31:105_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2"><i><font color="Green">E95</font></i></a></i>;<br/><a NAME="E31:105_1_2_7_1_3_3"/>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/><a NAME="E32:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E440</font></i>: 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E12:105_1_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><a NAME="E33:105_1_2_7_1_3_3"/><i><font color="Green">E441</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><a NAME="E34:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E442</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E20:105_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/><a NAME="E35:105_1_2_7_1_3_3"/><i><font color="Green">E443</font></i>: 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">DG</font>
 <b>by </b><i>, <a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/><a NAME="E36:105_1_2_7_1_3_3"/><b>then </b>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="mesfunc5.html#T41" target="_self">Th41</a></i>;<br/><a NAME="E37:105_1_2_7_1_3_3"/><b>then </b>
<font color="Maroon">e</font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="extreal1.html#K2">*</a> <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E31:105_1_2"><i><font color="Green">E134</font></i></a>, , <a class="ref" href="extreal1.html#T44">EXTREAL1:44</a></i>;<br/><a NAME="E38:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E444</font></i>: 
(  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">EP</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">EP</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T43" target="_self">Th43</a>, <a class="txt" href="mesfunc5.html#E31:105_1_2"><i><font color="Green">E134</font></i></a>, <a class="ref" href="extreal2.html#T45">EXTREAL2:45</a>, <a class="ref" href="xxreal_0.html#T4">XXREAL_0:4</a></i>;<br/><a NAME="E39:105_1_2_7_1_3_3"/><i><font color="Green">E445</font></i>: 
( <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E40:105_1_2_7_1_3_3"/><i><font color="Green">E446</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E41:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E447</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E42:105_1_2_7_1_3_3"/><i><font color="Green">E448</font></i>: 
( <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E28:105_1_2"><i><font color="Green">E129</font></i></a>, </i>;<br/><a NAME="E43:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E449</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="mesfunc5.html#T24" target="_self">Th24</a>, </i>;<br/><a NAME="E44:105_1_2_7_1_3_3"/><i><font color="Green">E450</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="ref" href="mesfunc5.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E45:105_1_2_7_1_3_3"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">EP</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">EP</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E31:105_1_2"><i><font color="Green">E134</font></i></a></i>;<br/><a NAME="E46:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E451</font></i>: 
<font color="Maroon">EP</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#T71">SUPINF_2:71</a></i>;<br/><b>then </b><i><font color="Green">E452</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">EP</font></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<font color="Maroon">B</font>
<b>by </b><i>, </i>
;<br/><a NAME="E48:105_1_2_7_1_3_3"/><i><font color="Green">E481</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i>, , </i>;<br/><a NAME="E49:105_1_2_7_1_3_3"/><i><font color="Green">E482</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_3_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_1_3_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:105_1_2_7_1_3_3_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E3:105_1_2_7_1_3_3_2"/><b>then </b><i><font color="Green">E483</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:105_1_2_7_1_3_3_2"/><b>then </b><i><font color="Green">E484</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:105_1_2_7_1_3_3_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">EP</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D3">MESFUNC1:def 3</a></i>;<br/>
<a NAME="E6:105_1_2_7_1_3_3_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">EP</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E7:105_1_2_7_1_3_3_2"/><b>then </b><i><font color="Green">E485</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">e</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:105_1_2_7_1_3_3_2"/><i><font color="Green">E486</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>set </b><font color="Maroon">f</font> = <font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span>;<br/>
<a NAME="E9:105_1_2_7_1_3_3_2"/><i><font color="Green">E487</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E10:105_1_2_7_1_3_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">e</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E11:105_1_2_7_1_3_3_2"/><b>then </b><i><font color="Green">E488</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp;  ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> ) )
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/>
<a NAME="E12:105_1_2_7_1_3_3_2"/><b>then </b><i><font color="Green">E489</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E13:105_1_2_7_1_3_3_2"/>
( <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E14:105_1_2_7_1_3_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , <a class="ref" href="extreal2.html#T33">EXTREAL2:33</a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E50:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E490</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i>, , , , </i>;<br/><a NAME="E51:105_1_2_7_1_3_3"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">EP</font></span>)</span>
 <b>by </b><i>, , , , </i>;<br/><a NAME="E52:105_1_2_7_1_3_3"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="grfunc_1.html#T64">GRFUNC_1:64</a>, , , , </i>;<br/><a NAME="E53:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E491</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E1:105_1_2"><i><font color="Green">E94</font></i></a>, , , , </i>;<br/> <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">EP</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">EP</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span>
<b>by </b><i>, , , , , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">EP</font></span>)</span>
<b>by </b><i>, <a class="ref" href="grfunc_1.html#T64">GRFUNC_1:64</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">EP</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>
;<br/><a NAME="E55:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E492</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">EP</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal2.html#T34">EXTREAL2:34</a>, <a class="txt" href="mesfunc5.html#E54"><i><font color="Green">Lemma172</font></i></a>, </i>;<br/><a NAME="E56:105_1_2_7_1_3_3"/><i><font color="Green">E493</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E57:105_1_2_7_1_3_3"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 ;<br/><a NAME="E58:105_1_2_7_1_3_3"/><b>then </b>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E59:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E494</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="txt" href="mesfunc5.html#E28:105_1_2"><i><font color="Green">E129</font></i></a></i>;<br/><b>consider </b><font color="Maroon">GP</font> being   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E61:105_1_2_7_1_3_3"/><i><font color="Green">E495</font></i>: 
( <font color="Maroon">GP</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">GP</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">GP</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span> ) )
 <b>by </b><i/>;<br/><a NAME="E62:105_1_2_7_1_3_3"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">GP</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">GP</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green">E130</font></i></a></i>;<br/><a NAME="E63:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E496</font></i>: 
<font color="Maroon">GP</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#T71">SUPINF_2:71</a></i>;<br/><a NAME="E64:105_1_2_7_1_3_3"/><i><font color="Green">E497</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">GP</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">GP</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_3_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_1_3_3_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">GP</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:105_1_2_7_1_3_3_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">GP</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">GP</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">GP</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E3:105_1_2_7_1_3_3_4"/><b>then </b><i><font color="Green">E498</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">GP</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:105_1_2_7_1_3_3_4"/><b>then </b><i><font color="Green">E499</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">GP</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:105_1_2_7_1_3_3_4"/><i><font color="Green">E500</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E6:105_1_2_7_1_3_3_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E7:105_1_2_7_1_3_3_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:105_1_2_7_1_3_3_4"/><b>then </b>
<font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2"><i><font color="Green">E124</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:105_1_2_7_1_3_3_4"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">GP</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><a NAME="E65:105_1_2_7_1_3_3"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">GP</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">GP</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 <b>by </b><i>, , , , </i>;<br/><a NAME="E66:105_1_2_7_1_3_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">GP</font> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font> &amp; <font color="Maroon">GP</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">GP</font> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">GP</font> )
 <b>by </b><i><a class="ref" href="grfunc_1.html#T64">GRFUNC_1:64</a>, , </i>;<br/><a NAME="E67:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E501</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">GP</font>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E1:105_1_2"><i><font color="Green">E94</font></i></a>, , , , </i>;<br/><a NAME="E68:105_1_2_7_1_3_3"/><i><font color="Green">E502</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">GP</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green">E130</font></i></a>, , <a class="txt" href="mesfunc5.html#E4:105_1_2"><i><font color="Green">E95</font></i></a></i>;<br/><a NAME="E69:105_1_2_7_1_3_3"/><i><font color="Green">E503</font></i>: 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="mesfunc5.html#T41" target="_self">Th41</a>, , <a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/><a NAME="E70:105_1_2_7_1_3_3"/>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">DG</font>
 <b>by </b><i>, <a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/><a NAME="E71:105_1_2_7_1_3_3"/><b>then </b>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">A</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="mesfunc5.html#T41" target="_self">Th41</a></i>;<br/><a NAME="E72:105_1_2_7_1_3_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="extreal1.html#T44">EXTREAL1:44</a></i>;<br/><a NAME="E73:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E504</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>;<br/><a NAME="E74:105_1_2_7_1_3_3"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><a NAME="E75:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E505</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E20:105_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/><a NAME="E76:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E506</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="extreal2.html#T45">EXTREAL2:45</a></i>;<br/><a NAME="E77:105_1_2_7_1_3_3"/>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">DG</font> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, </i>;<br/><a NAME="E78:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E507</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="extreal1.html#T44">EXTREAL1:44</a></i>;<br/><a NAME="E79:105_1_2_7_1_3_3"/><b>then </b>
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <font color="Maroon">e</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E80:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E508</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T4">XXREAL_0:4</a></i>;<br/><a NAME="E81:105_1_2_7_1_3_3"/>
( <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E31:105_1_2"><i><font color="Green">E134</font></i></a>, <a class="ref" href="xxreal_0.html#T4">XXREAL_0:4</a></i>;<br/><b>then reconsider </b><font color="Maroon">ee</font> = <font color="Maroon">e</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><a NAME="E83:105_1_2_7_1_3_3"/><i><font color="Green">E509</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">beta1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">ee</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T13">EXTREAL1:13</a></i>;<br/><a NAME="E84:105_1_2_7_1_3_3"/><i><font color="Green">E510</font></i>: 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E20:105_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="mesfunc5.html#T42" target="_self">Th42</a>, <a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/><a NAME="E85:105_1_2_7_1_3_3"/>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E53"><i><font color="Green">Lemma171</font></i></a>, <a class="ref" href="mesfunc5.html#T42" target="_self">Th42</a>, <a class="txt" href="mesfunc5.html#E20:105_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/><a NAME="E86:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E511</font></i>: 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E12:105_1_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><b>reconsider </b><font color="Maroon">XSMg</font> =  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><b>reconsider </b><font color="Maroon">XSMgA</font> =  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, , <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><b>reconsider </b><font color="Maroon">XSMGP</font> =  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">GP</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, , , <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><b>reconsider </b><font color="Maroon">MA</font> = <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">A</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, , <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><b>reconsider </b><font color="Maroon">betae</font> = <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <font color="Maroon">e</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">XSMEP</font> =  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">EP</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="mesfunc5.html#E54"><i><font color="Green">Lemma172</font></i></a>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><a NAME="E93:105_1_2_7_1_3_3"/>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B</font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E53"><i><font color="Green">Lemma171</font></i></a>, , <a class="ref" href="mesfunc5.html#T41" target="_self">Th41</a>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><b>then consider </b><font color="Maroon">MB</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">MG</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E95:105_1_2_7_1_3_3"/><i><font color="Green">E519</font></i>: 
( <font color="Maroon">MB</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B</font> &amp; <font color="Maroon">MG</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> &amp; <font color="Maroon">MB</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">MG</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E53"><i><font color="Green">Lemma171</font></i></a>, </i>;<br/><i><font color="Green">E520</font></i>: <font color="Maroon">XSMEP</font> = 
<font color="Maroon">e</font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E31:105_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2"><i><font color="Green">E95</font></i></a></i>
<br/>.= 
<font color="Maroon">ee</font> <a href="real_1.html#K4">*</a> <font color="Maroon">MB</font>
<b>by </b><i>, <a class="ref" href="extreal1.html#T13">EXTREAL1:13</a></i>
;<br/><a NAME="E97:105_1_2_7_1_3_3"/><i><font color="Green">E521</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="extreal2.html#T11">EXTREAL2:11</a></i>;<br/><a NAME="E98:105_1_2_7_1_3_3"/><i><font color="Green">E522</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">GP</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a>, , , , </i>;<br/><font color="Maroon">XSMg</font> <a href="real_1.html#K5">-</a> <font color="Maroon">XSMgA</font> = 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>
<br/>.= 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="extreal2.html#T11">EXTREAL2:11</a></i>
;<br/><b>then reconsider </b><font color="Maroon">XSMgB</font> =  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E101:105_1_2_7_1_3_3"/><i><font color="Green">E524</font></i>: 
<font color="Maroon">XSMg</font> <a href="real_1.html#K5">-</a> <font color="Maroon">betae</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">XSMg</font> <a href="real_1.html#K5">-</a> <font color="Maroon">XSMGP</font>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a>, , </i>;<br/><a NAME="E102:105_1_2_7_1_3_3"/>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">GP</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">XSMg</font> <a href="real_1.html#K5">-</a> <font color="Maroon">XSMGP</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/><a NAME="E103:105_1_2_7_1_3_3"/><b>then </b>
<font color="Maroon">XSMg</font> <a href="real_1.html#K5">-</a> <font color="Maroon">betae</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">XSMgB</font>
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E104:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E525</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">XSMg</font> <a href="real_1.html#K5">-</a> <font color="Maroon">betae</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">ee</font> <a href="real_1.html#K4">*</a> <font color="Maroon">MB</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">XSMgB</font> <a href="real_1.html#K5">-</a> <font color="Maroon">XSMEP</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E105:105_1_2_7_1_3_3"/><i><font color="Green">E526</font></i>: 
<font color="Maroon">betae</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ee</font> <a href="real_1.html#K4">*</a> <font color="Maroon">beta1</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T13">EXTREAL1:13</a></i>;<br/><a NAME="E106:105_1_2_7_1_3_3"/>
<font color="Maroon">ee</font> <a href="real_1.html#K4">*</a> <font color="Maroon">MB</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ee</font> <a href="real_1.html#K4">*</a> <font color="Maroon">MG</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E31:105_1_2"><i><font color="Green">E134</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a>, </i>;<br/><a NAME="E107:105_1_2_7_1_3_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">XSMg</font> <a href="real_1.html#K5">-</a> <font color="Maroon">betae</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">ee</font> <a href="real_1.html#K4">*</a> <font color="Maroon">MG</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">XSMg</font> <a href="real_1.html#K5">-</a> <font color="Maroon">betae</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">ee</font> <a href="real_1.html#K4">*</a> <font color="Maroon">MB</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T15">XREAL_1:15</a></i>;<br/><a NAME="E108:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E527</font></i>: 
<font color="Maroon">XSMg</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">ee</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">MG</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">XSMgB</font> <a href="real_1.html#K5">-</a> <font color="Maroon">XSMEP</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E109:105_1_2_7_1_3_3"/>
<font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">MG</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/><a NAME="E110:105_1_2_7_1_3_3"/><b>then </b>
<font color="Maroon">ee</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">MG</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T13">EXTREAL1:13</a></i>;<br/><a NAME="E111:105_1_2_7_1_3_3"/><b>then </b><i><font color="Green">E528</font></i>: 
<font color="Maroon">XSMg</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">ee</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">MG</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/><a NAME="E112:105_1_2_7_1_3_3"/>
<font color="Maroon">XSMgB</font> <a href="real_1.html#K5">-</a> <font color="Maroon">XSMEP</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">EP</font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/><a NAME="E113:105_1_2_7_1_3_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E114:105_1_2_7_1_3_3"/>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T40" target="_self">Th40</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E29:105_1_2_7_1_3"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E16:105_1_2_7_1"/><i><font color="Green">E529</font></i>: 
for <font color="Olive">e</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> &amp; <font color="Olive">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span> holds <br/> ex <font color="Olive">N0</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Olive">e</font> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_1_4"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> &amp; <font color="Maroon">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span> )
 ;<br/>

<b>then consider </b><font color="Maroon">N0</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:105_1_2_7_1_4"/><i><font color="Green">E530</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span>
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:105_1_2_7_1_4_1"/>
<font color="Maroon">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:105_1_2_7_1_4_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:105_1_2_7_1_4_1"/>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:105_1_2_7_1_4"/>
 ex <font color="Olive">N0</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E17:105_1_2_7_1"/><i><font color="Green">E531</font></i>: 
for <font color="Olive">e1</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e1</font> holds <br/> ex <font color="Olive">e</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> &amp; <font color="Olive">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span> &amp; <font color="Olive">e</font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">e1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e1</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_1_5"/><i><font color="Green">E532</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">ralpha</font> =  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">rdomg</font> = <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="mesfunc5.html#E24:105_1_2"><i><font color="Green">E128</font></i></a>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/>
<a NAME="E4:105_1_2_7_1_5"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">DG</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E5:105_1_2_7_1_5"/><b>then </b><i><font color="Green">E535</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">rdomg</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E20:105_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7_1_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1_2_7_1_5_1"/>
( <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">e1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7_1_5_1_1"><b>suppose </b></a><a NAME="E1:105_1_2_7_1_5_1_1"/><i><font color="Green">E536</font></i>: 
<font color="Maroon">e1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon">z</font> = <font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2;<br/><a NAME="E2:105_1_2_7_1_5_1_1"/><i><font color="Green">E537</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2 &amp; <font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T217">XREAL_1:217</a>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/><a NAME="E3:105_1_2_7_1_5_1_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/><b>hence </b><a NAME="E4:105_1_2_7_1_5_1_1"/>
 ex <font color="Olive">e</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> &amp; <font color="Olive">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span> &amp; <font color="Olive">e</font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e1</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7_1_5_1_2"><b>suppose </b></a><a NAME="E1:105_1_2_7_1_5_1_2"/>
<font color="Maroon">e1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">re1</font> = <font color="Maroon">e1</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><a NAME="E3:105_1_2_7_1_5_1_2"/><i><font color="Green">E539</font></i>: 
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>set </b><font color="Maroon">x</font> = <font color="Maroon">re1</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span>;<br/><b>set </b><font color="Maroon">y</font> = <font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2;<br/><a NAME="E4:105_1_2_7_1_5_1_2"/><i><font color="Green">E540</font></i>: 
<font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ralpha</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/><b>set </b><font color="Maroon">z</font> =  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">re1</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span>;<br/><div><i><font color="Green">E541</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_5_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7_1_5_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1_2_7_1_5_1_2_1_1"/>
(  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">re1</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">re1</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span> or  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">re1</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2 )
 </span><b>by </b><i><a class="ref" href="square_1.html#T38">SQUARE_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7_1_5_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:105_1_2_7_1_5_1_2_1_1_1"/>
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">re1</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">re1</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:105_1_2_7_1_5_1_2_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">re1</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7_1_5_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:105_1_2_7_1_5_1_2_1_1_2"/>
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">re1</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:105_1_2_7_1_5_1_2_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">re1</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T217">XREAL_1:217</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E6:105_1_2_7_1_5_1_2"/>
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">re1</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="square_1.html#T35">SQUARE_1:35</a></i>;<br/><a NAME="E7:105_1_2_7_1_5_1_2"/><b>then </b><i><font color="Green">E542</font></i>: 
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">re1</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ralpha</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E8:105_1_2_7_1_5_1_2"/>
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">re1</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">re1</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span>
 <b>by </b><i><a class="ref" href="square_1.html#T35">SQUARE_1:35</a></i>;<br/><a NAME="E9:105_1_2_7_1_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><font color="Maroon">re1</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">re1</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E10:105_1_2_7_1_5_1_2"/><b>then </b><i><font color="Green">E543</font></i>: 
<span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><font color="Maroon">re1</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">re1</font>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E11:105_1_2_7_1_5_1_2"/>
 <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">beta1</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">rdomg</font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/><a NAME="E12:105_1_2_7_1_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><font color="Maroon">re1</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p3">(<span class="default"><font color="Maroon">re1</font> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">beta1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">rdomg</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T38">EXTREAL1:38</a></i>;<br/><b>hence </b><a NAME="E13:105_1_2_7_1_5_1_2"/>
 ex <font color="Olive">e</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> &amp; <font color="Olive">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span> &amp; <font color="Olive">e</font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e1</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E18:105_1_2_7_1"/><i><font color="Green">E544</font></i>: 
for <font color="Olive">e1</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e1</font> holds <br/> ex <font color="Olive">N0</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Olive">e1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e1</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_1_6"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">e</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:105_1_2_7_1_6"/><i><font color="Green">E545</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> &amp; <font color="Maroon">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span> &amp; <font color="Maroon">e</font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e1</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">N0</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:105_1_2_7_1_6"/><i><font color="Green">E546</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E31:105_1_2"><i><font color="Green">E134</font></i></a>, </i>;<br/>
<b>take </b>
<font color="Maroon">N0</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:105_1_2_7_1_6_1"/>
<font color="Maroon">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:105_1_2_7_1_6_1"/><b>then </b><i><font color="Green">E547</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E3:105_1_2_7_1_6_1"/>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">e1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a>, , </i>;<br/><b>hence </b><a NAME="E4:105_1_2_7_1_6_1"/>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">e1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:105_1_2_7_1_6"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">e1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E19:105_1_2_7_1"/><i><font color="Green">E548</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_1_7"/><i><font color="Green">E549</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:105_1_2_7_1_7"/><i><font color="Green">E550</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:105_1_2_7_1_7"/><i><font color="Green">E551</font></i>: 
( <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E6:105_1_2"><i><font color="Green">E103</font></i></a></i>;<br/>
<a NAME="E4:105_1_2_7_1_7"/><i><font color="Green">E552</font></i>: 
( <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:105_1_2_7_1_7"/><i><font color="Green">E553</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_1_7_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:105_1_2_7_1_7_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E3:105_1_2_7_1_7_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:105_1_2_7_1_7_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:105_1_2_7_1_7"/><b>then </b><i><font color="Green">E554</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2"><i><font color="Green">E94</font></i></a>, , </i>;<br/>
<a NAME="E7:105_1_2_7_1_7"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E8:105_1_2_7_1_7"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T64">GRFUNC_1:64</a>, </i>;<br/>
<a NAME="E9:105_1_2_7_1_7"/><b>then </b>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E10:105_1_2_7_1_7"/>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E20:105_1_2_7_1"/><i><font color="Green">E555</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:105_1_2_7_1_8"/><i><font color="Green">E556</font></i>: 
<font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E6:105_1_2"><i><font color="Green">E103</font></i></a></i>;<br/>
<a NAME="E2:105_1_2_7_1_8"/>
<font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:105_1_2_7_1_8"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T24" target="_self">Th24</a>, </i>;<br/>
<b>hence </b><a NAME="E4:105_1_2_7_1_8"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7_1_9"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1_2_7_1_9"/>
(  ex <font color="Olive">K</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">K</font> &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  holds <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">K</font> ) ) or for <font color="Olive">K</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   holds <br/>(  not 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">K</font> or  ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> st not <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">K</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7_1_9_1"><b>suppose </b></a><a NAME="E1:105_1_2_7_1_9_1"/>
 ex <font color="Olive">K</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">K</font> &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  holds <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">K</font> ) )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">K</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:105_1_2_7_1_9_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E4:105_1_2_7_1_9_1"/><i><font color="Green">E557</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  holds <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E5:105_1_2_7_1_9_1"/><i><font color="Green">E558</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_9_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:105_1_2_7_1_9_1_1"/><i><font color="Green">E559</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span> is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:105_1_2_7_1_9_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:105_1_2_7_1_9_1_1"/><b>then </b>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:105_1_2_7_1_9_1_1"/>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:105_1_2_7_1_9_1"/><b>then </b>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span>
 ;<br/><a NAME="E7:105_1_2_7_1_9_1"/><b>then </b><i><font color="Green">E560</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i/>;<br/><a NAME="E8:105_1_2_7_1_9_1"/>
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">K</font> is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_9_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_9_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>assume </b><a NAME="E1:105_1_2_7_1_9_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font>
 ;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:105_1_2_7_1_9_1_2_1"/><i><font color="Green">E561</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E4:105_1_2_7_1_9_1_2_1"/><i><font color="Green">E562</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:105_1_2_7_1_9_1_2_1"/>
<font color="Maroon">z</font> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E6:105_1_2_7_1_9_1_2_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">K</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:105_1_2_7_1_9_1_2"/>
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">K</font> is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:105_1_2_7_1_9_1"/><b>then </b><i><font color="Green">E563</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a></i>;<br/><a NAME="E10:105_1_2_7_1_9_1"/>
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">K</font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E11:105_1_2_7_1_9_1"/><b>then </b><i><font color="Green">E564</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/><a NAME="E12:105_1_2_7_1_9_1"/><i><font color="Green">E565</font></i>: 
for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> holds <br/> ex <font color="Olive">N0</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p4">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_9_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_1_9_1_3"/><i><font color="Green">E566</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:105_1_2_7_1_9_1_3"/>
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i>, , <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E4:105_1_2_7_1_9_1_3"/><i><font color="Green">E567</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font> &amp; <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="measure6.html#T33">MEASURE6:33</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:105_1_2_7_1_9_1_3"/><i><font color="Green">E568</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">L</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">N0</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a></i>;<br/>
<b>take </b>
<font color="Maroon">N0</font>
;<br/>

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E8:105_1_2_7_1_9_1_3"/>
<font color="Maroon">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E9:105_1_2_7_1_9_1_3"/><b>then </b>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:105_1_2_7_1_9_1_3"/><b>then </b><i><font color="Green">E569</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a>, <a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:105_1_2_7_1_9_1_3"/><i><font color="Green">E570</font></i>: 
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:105_1_2_7_1_9_1_3"/>
<span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a>, </i>;<br/>
<a NAME="E13:105_1_2_7_1_9_1_3"/><b>then </b><i><font color="Green">E571</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/>
<a NAME="E14:105_1_2_7_1_9_1_3"/>
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_9_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:105_1_2_7_1_9_1_3_1"/><i><font color="Green">E572</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span>
 ;<br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p3">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p4">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="extreal1.html#T8">EXTREAL1:8</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="extreal1.html#T9">EXTREAL1:9</a></i>
<br/>.= 
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
;<br/>
<b>hence </b><a NAME="E3:105_1_2_7_1_9_1_3_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="extreal1.html#T9">EXTREAL1:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:105_1_2_7_1_9_1_3"/><b>then </b>
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/>
<a NAME="E16:105_1_2_7_1_9_1_3"/><b>then </b>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E17:105_1_2_7_1_9_1_3"/><b>then </b><i><font color="Green">E573</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="extreal2.html#T30">EXTREAL2:30</a></i>;<br/>
<a NAME="E18:105_1_2_7_1_9_1_3"/>
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a>, <a class="ref" href="extreal2.html#T29">EXTREAL2:29</a></i>;<br/>
<a NAME="E19:105_1_2_7_1_9_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="extreal2.html#T30">EXTREAL2:30</a></i>;<br/>
<a NAME="E20:105_1_2_7_1_9_1_3"/><b>then </b>
 <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T17">SUPINF_2:17</a></i>;<br/>
<a NAME="E21:105_1_2_7_1_9_1_3"/><b>then </b>
 <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="extreal2.html#T15">EXTREAL2:15</a></i>;<br/>
<b>hence </b><a NAME="E22:105_1_2_7_1_9_1_3"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p4">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="extreal2.html#T59">EXTREAL2:59</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">h</font> =  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i>, , <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><a NAME="E14:105_1_2_7_1_9_1"/><i><font color="Green">E574</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span>
 ;<br/><a NAME="E15:105_1_2_7_1_9_1"/><b>then </b><i><font color="Green">E575</font></i>: 
<font color="Maroon">L</font> is <a href="mesfunc5.html#V7" target="_self">convergent_to_finite_number</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T11" target="_self">Th11</a>, <a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a></i>;<br/><b>hence </b><a NAME="E16:105_1_2_7_1_9_1"/>
<font color="Maroon">L</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E17:105_1_2_7_1_9_1"/><b>then </b><i><font color="Green">E576</font></i>: 
 <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a>, , <a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a>, <a class="ref" href="mesfunc5.html#T14" target="_self">Th14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_9_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:105_1_2_7_1_9_1_4"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 ;<br/><b>then consider </b><font color="Maroon">N0</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:105_1_2_7_1_9_1_4"/><i><font color="Green">E577</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">e</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font>
 <b>by </b><i/>;<br/><a NAME="E4:105_1_2_7_1_9_1_4"/><i><font color="Green">E578</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">e</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">N0</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a></i>;<br/><a NAME="E5:105_1_2_7_1_9_1_4"/>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">N0</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:105_1_2_7_1_9_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">e</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E7:105_1_2_7_1_9_1_4"/>
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">e</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a>, <a class="ref" href="extreal2.html#T29">EXTREAL2:29</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E19:105_1_2_7_1_9_1"/>
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7_1_9_2"><b>suppose </b></a><a NAME="E1:105_1_2_7_1_9_2"/><i><font color="Green">E579</font></i>: 
for <font color="Olive">K</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   holds <br/>(  not 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">K</font> or  ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> st not <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">K</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_9_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">K</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:105_1_2_7_1_9_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">K</font>
 ;<br/><b>then consider </b><font color="Maroon">N0</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:105_1_2_7_1_9_2_1"/><i><font color="Green">E580</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">K</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">N0</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_1_9_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:105_1_2_7_1_9_2_1_1"/>
<font color="Maroon">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:105_1_2_7_1_9_2_1_1"/><b>then </b>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:105_1_2_7_1_9_2_1_1"/>
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">K</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:105_1_2_7_1_9_2_1"/>
 ex <font color="Olive">N0</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><font color="Maroon">K</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font>
 ;<br/></div><b>end;</b></div><a NAME="E3:105_1_2_7_1_9_2"/><b>then </b><i><font color="Green">E581</font></i>: 
<font color="Maroon">L</font> is <a href="mesfunc5.html#V8" target="_self">convergent_to_+infty</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T13" target="_self">Th13</a></i>;<br/><b>hence </b><a NAME="E4:105_1_2_7_1_9_2"/>
<font color="Maroon">L</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E5:105_1_2_7_1_9_2"/><b>then </b>
 <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T14" target="_self">Th14</a></i>;<br/><b>hence </b><a NAME="E6:105_1_2_7_1_9_2"/>
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_7_2"><b>suppose </b></a><a NAME="E1:105_1_2_7_2"/><i><font color="Green">E582</font></i>: 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">DG</font> =  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i>, </i>;<br/><a NAME="E3:105_1_2_7_2"/><i><font color="Green">E583</font></i>: 
for <font color="Olive">e</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> &amp; <font color="Olive">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span> holds <br/>for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Olive">e</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span></span>)</span>,<font color="Olive">e</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_2_1"/><i><font color="Green">E584</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> &amp; <font color="Maroon">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span> )
 ;<br/>

<a NAME="E2:105_1_2_7_2_1"/><b>then </b><i><font color="Green">E585</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T4">XXREAL_0:4</a></i>;<br/>
<a NAME="E3:105_1_2_7_2_1"/>
<font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/>
<a NAME="E4:105_1_2_7_2_1"/><b>then </b>
<font color="Maroon">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E24:105_1_2"><i><font color="Green">E128</font></i></a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/>
<a NAME="E5:105_1_2_7_2_1"/><b>then </b><i><font color="Green">E586</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="extreal1.html#T40">EXTREAL1:40</a></i>;<br/>
<b>consider </b><font color="Maroon">H</font> being   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>, <font color="Maroon">MF</font> being   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E7:105_1_2_7_2_1"/><i><font color="Green">E587</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span></span>)</span>,<font color="Maroon">e</font>
 <b>and </b><br/><a NAME="E8:105_1_2_7_2_1"/>
for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font>
 <b>and </b><br/><a NAME="E9:105_1_2_7_2_1"/><i><font color="Green">E588</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E10:105_1_2_7_2_1"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">MF</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span>
 <b>and </b><br/><a NAME="E11:105_1_2_7_2_1"/>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">MF</font></span>)</span>
 <b>and </b><br/><a NAME="E12:105_1_2_7_2_1"/><i><font color="Green">E589</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:105_1_2_7_2_1_1"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">DG</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green">E131</font></i></a></i>;<br/><b>then </b><i><font color="Green">E590</font></i>: <font color="Maroon">DG</font> = 
<font color="Maroon">DG</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">DG</font> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
;<br/><a NAME="E3:105_1_2_7_2_1_1"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font>
 <b>by </b><i/>;<br/><a NAME="E4:105_1_2_7_2_1_1"/><b>then </b>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="measure1.html#R2">is_measurable</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#D12">MEASURE1:def 12</a></i>;<br/><a NAME="E5:105_1_2_7_2_1_1"/><b>then </b><i><font color="Green">E591</font></i>: 
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure1.html#R2">is_measurable</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#T66">MEASURE1:66</a></i>;<br/><a NAME="E6:105_1_2_7_2_1_1"/>
<font color="Maroon">DG</font> <a href="measure1.html#R2">is_measurable</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#D12">MEASURE1:def 12</a></i>;<br/><a NAME="E7:105_1_2_7_2_1_1"/><b>then </b><i><font color="Green">E592</font></i>: 
<font color="Maroon">DG</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="measure1.html#R2">is_measurable</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#T66">MEASURE1:66</a>, <a class="txt" href="mesfunc5.html#E31:105_1_2"><i><font color="Green">E134</font></i></a></i>;<br/><font color="Maroon">DG</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">DG</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T49">XBOOLE_1:49</a></i>
<br/>.= 
<font color="Maroon">DG</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">DG</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i>, <a class="ref" href="measure1.html#D12">MEASURE1:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/><a NAME="E11:105_1_2_7_2_1_1"/><i><font color="Green">E593</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/><a NAME="E12:105_1_2_7_2_1_1"/><b>then </b><i><font color="Green">E594</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E13:105_1_2_7_2_1_1"/><i><font color="Green">E595</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E14:105_1_2_7_2_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 ;<br/><a NAME="E15:105_1_2_7_2_1_1"/><b>then </b><i><font color="Green">E596</font></i>: 
<font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E16:105_1_2_7_2_1_1"/><i><font color="Green">E597</font></i>: 
<font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E6:105_1_2"><i><font color="Green">E103</font></i></a></i>;<br/><a NAME="E17:105_1_2_7_2_1_1"/><i><font color="Green">E598</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><a NAME="E18:105_1_2_7_2_1_1"/><i><font color="Green">E599</font></i>: 
<font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/><a NAME="E19:105_1_2_7_2_1_1"/><b>then </b><i><font color="Green">E600</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i>, , , </i>;<br/><a NAME="E20:105_1_2_7_2_1_1"/><i><font color="Green">E601</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 <b>by </b><i>, </i>;<br/><a NAME="E21:105_1_2_7_2_1_1"/><i><font color="Green">E602</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E22:105_1_2_7_2_1_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T24" target="_self">Th24</a>, <a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E23:105_1_2_7_2_1_1"/><b>then </b><i><font color="Green">E603</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T20">SUPINF_2:20</a></i>;<br/><i><font color="Green">E604</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<font color="Maroon">B</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><a NAME="E25:105_1_2_7_2_1_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="supinf_2.html#T10">SUPINF_2:10</a></i>;<br/><b>then consider </b><font color="Maroon">EP</font> being   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E27:105_1_2_7_2_1_1"/><i><font color="Green">E605</font></i>: 
( <font color="Maroon">EP</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">EP</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">EP</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">e</font> ) )
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E28:105_1_2"><i><font color="Green">E129</font></i></a></i>;<br/><a NAME="E28:105_1_2_7_2_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">EP</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">EP</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a>, <a class="txt" href="mesfunc5.html#E28:105_1_2"><i><font color="Green">E129</font></i></a></i>;<br/><a NAME="E29:105_1_2_7_2_1_1"/><b>then </b><i><font color="Green">E606</font></i>: 
<font color="Maroon">EP</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#T71">SUPINF_2:71</a></i>;<br/><a NAME="E30:105_1_2_7_2_1_1"/><i><font color="Green">E607</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">EP</font></span>)</span> holds <br/><font color="Maroon">EP</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_2_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_2_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">EP</font></span>)</span>
 ;<br/>

<a NAME="E2:105_1_2_7_2_1_1_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">EP</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">EP</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">EP</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E3:105_1_2_7_2_1_1_4"/><b>then </b><i><font color="Green">E608</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">EP</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:105_1_2_7_2_1_1_4"/><b>then </b><i><font color="Green">E609</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:105_1_2_7_2_1_1_4"/><b>then </b><i><font color="Green">E610</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>set </b><font color="Maroon">f</font> = <font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span>;<br/>
<a NAME="E6:105_1_2_7_2_1_1_4"/><i><font color="Green">E611</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E7:105_1_2_7_2_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">e</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green">E130</font></i></a>, , <a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E8:105_1_2_7_2_1_1_4"/><b>then </b><i><font color="Green">E612</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp;  ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> ) )
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/>
<a NAME="E9:105_1_2_7_2_1_1_4"/><b>then </b><i><font color="Green">E613</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E10:105_1_2_7_2_1_1_4"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/>( <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_2_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_2_1_1_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:105_1_2_7_2_1_1_4_1"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T30" target="_self">Th30</a>, <a class="ref" href="mesfunc2.html#D1">MESFUNC2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:105_1_2_7_2_1_1_4_1"/>
( <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="ref" href="extreal2.html#T67">EXTREAL2:67</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:105_1_2_7_2_1_1_4"/><b>then </b>
( <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T37" target="_self">Th37</a>, <a class="ref" href="mesfunc5.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E12:105_1_2_7_2_1_1_4"/><b>then </b>
<font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">e</font>
 
<b>by </b><i>, <a class="ref" href="extreal2.html#T33">EXTREAL2:33</a>, <a class="txt" href="mesfunc5.html#E24:105_1_2"><i><font color="Green">E128</font></i></a></i>;<br/>
<a NAME="E13:105_1_2_7_2_1_1_4"/><b>then </b><i><font color="Green">E614</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E24:105_1_2"><i><font color="Green">E128</font></i></a>, <a class="ref" href="extreal2.html#T34">EXTREAL2:34</a></i>;<br/>
<a NAME="E14:105_1_2_7_2_1_1_4"/>
 <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E10:105_1_2"><i><font color="Green">E124</font></i></a>, <a class="ref" href="mesfunc5.html#T36" target="_self">Th36</a>, <a class="ref" href="mesfunc5.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E15:105_1_2_7_2_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a>, <a class="txt" href="mesfunc5.html#E24:105_1_2"><i><font color="Green">E128</font></i></a></i>;<br/>
<a NAME="E16:105_1_2_7_2_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E46"><i><font color="Green">Lemma157</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E17:105_1_2_7_2_1_1_4"/>
<font color="Maroon">EP</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a>, <a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a></i>;<br/>


</div><b>end;</b></div><a NAME="E31:105_1_2_7_2_1_1"/><b>then </b><i><font color="Green">E615</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">EP</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">EP</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">EP</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:105_1_2"><i><font color="Green">E94</font></i></a>, , <a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a>, <a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a>, </i>;<br/><a NAME="E32:105_1_2_7_2_1_1"/><i><font color="Green">E616</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">EP</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">e</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a>, <a class="txt" href="mesfunc5.html#E28:105_1_2"><i><font color="Green">E129</font></i></a>, <a class="txt" href="mesfunc5.html#E4:105_1_2"><i><font color="Green">E95</font></i></a></i>;<br/><a NAME="E33:105_1_2_7_2_1_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">EP</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">EP</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a>, <a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a>, , , </i>;<br/><a NAME="E34:105_1_2_7_2_1_1"/><b>then </b>
( <font color="Maroon">EP</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">EP</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">EP</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">EP</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="grfunc_1.html#T64">GRFUNC_1:64</a>, <a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a>, <a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a></i>;<br/><a NAME="E35:105_1_2_7_2_1_1"/><b>then </b>
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">EP</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, </i>;<br/><b>hence </b><a NAME="E36:105_1_2_7_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">e</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">e</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green">E130</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:105_1_2_7_2_1"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">e</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span></span>)</span>,<font color="Maroon">e</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E4:105_1_2_7_2"/>
for <font color="Olive">y</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">y</font> holds <br/> ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_2_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">y</font>
 ;<br/>

<b>set </b><font color="Maroon">e</font> = <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">ralpha</font> =  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/>
<b>set </b><font color="Maroon">a2</font> = <font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2;<br/>
<a NAME="E3:105_1_2_7_2_2"/><i><font color="Green">E618</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="extreal1.html#T39">EXTREAL1:39</a></i>;<br/>
<a NAME="E4:105_1_2_7_2_2"/><b>then </b><i><font color="Green">E619</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T217">XREAL_1:217</a></i>;<br/>
<a NAME="E5:105_1_2_7_2_2"/><i><font color="Green">E620</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a>, <a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a></i>;<br/>
<a NAME="E6:105_1_2_7_2_2"/><i><font color="Green">E621</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">ralpha</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/>
<a NAME="E7:105_1_2_7_2_2"/><i><font color="Green">E622</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ralpha</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T16">XREAL_1:16</a></i>;<br/>
<a NAME="E8:105_1_2_7_2_2"/><b>then </b><i><font color="Green">E623</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E10:105_1_2_7_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a>, <a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green">E131</font></i></a></i>;<br/>
<a NAME="E11:105_1_2_7_2_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="extreal1.html#T38">EXTREAL1:38</a></i>;<br/>
<a NAME="E12:105_1_2_7_2_2"/><b>then </b><i><font color="Green">E624</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p5">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p4">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E29:105_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green">E131</font></i></a>, <a class="ref" href="extreal1.html#T39">EXTREAL1:39</a></i>;<br/>
<a NAME="E13:105_1_2_7_2_2"/><i><font color="Green">E625</font></i>: 
<font color="Maroon">y</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<font color="Maroon">y</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">y1</font></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">ralpha</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E30:105_1_2"><i><font color="Green">E131</font></i></a>, <a class="ref" href="extreal1.html#T39">EXTREAL1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>
;<br/>
<a NAME="E15:105_1_2_7_2_2"/><b>then </b><i><font color="Green">E626</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E31:105_1_2"><i><font color="Green">E134</font></i></a>, <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/>
<b>consider </b><font color="Maroon">H</font> being   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>, <font color="Maroon">MF</font> being   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E17:105_1_2_7_2_2"/><i><font color="Green">E627</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E18:105_1_2_7_2_2"/><i><font color="Green">E628</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font>
 <b>and </b><br/><a NAME="E19:105_1_2_7_2_2"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E20:105_1_2_7_2_2"/><i><font color="Green">E629</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">MF</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span>
 <b>and </b><br/><a NAME="E21:105_1_2_7_2_2"/><i><font color="Green">E630</font></i>: 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">MF</font></span>)</span>
 <b>and </b><br/><a NAME="E22:105_1_2_7_2_2"/><i><font color="Green">E631</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E24:105_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="mesfunc5.html#E28:105_1_2"><i><font color="Green">E129</font></i></a>, </i>;<br/>
<a NAME="E23:105_1_2_7_2_2"/>
 ex <font color="Olive">z</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">MF</font> &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_2_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:105_1_2_7_2_2_2"/>
for <font color="Olive">z</font> being  <a href="supinf_1.html#NM3">R_eal</a>  holds <br/>(  not <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">MF</font> or  not <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">z</font> )
 ;<br/>

<a NAME="E2:105_1_2_7_2_2_2"/><b>then </b>
for <font color="Olive">z</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">MF</font> holds <br/><font color="Olive">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E3:105_1_2_7_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">MF</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:105_1_2_7_2_2_2"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="mesfunc5.html#E12:105_1_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">z</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E25:105_1_2_7_2_2"/><i><font color="Green">E632</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <font color="Maroon">MF</font> &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font> )
 ;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E27:105_1_2_7_2_2"/><i><font color="Green">E633</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">MF</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MF</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">N0</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">N0</font>
;<br/>

<b>thus </b><a NAME="E29:105_1_2_7_2_2"/>
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_7_2_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:105_1_2_7_2_2_3"/><i><font color="Green">E634</font></i>: 
<font color="Maroon">N0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:105_1_2_7_2_2_3"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">N0</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:105_1_2_7_2_2_3"/><b>then </b><i><font color="Green">E635</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">N0</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="extreal1.html#T42">EXTREAL1:42</a></i>;<br/>
<a NAME="E4:105_1_2_7_2_2_3"/><i><font color="Green">E636</font></i>: 
<font color="Maroon">H</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">N0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:105_1_2_7_2_2_3"/>
( <font color="Maroon">H</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">N0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E6:105_1_2_7_2_2_3"/><b>then </b>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">N0</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/>
<a NAME="E7:105_1_2_7_2_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">N0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="extreal1.html#T42">EXTREAL1:42</a></i>;<br/>
<a NAME="E8:105_1_2_7_2_2_3"/><b>then </b>
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:105_1_2_7_2_2_3"/><b>then </b><i><font color="Green">E637</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p5">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p4">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E10:105_1_2_7_2_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p5">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">RINGA</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p4">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E20:105_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="mesfunc5.html#E24:105_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="mesfunc5.html#E28:105_1_2"><i><font color="Green">E129</font></i></a></i>;<br/>
<a NAME="E11:105_1_2_7_2_2_3"/><b>then </b>
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a>, <a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a></i>;<br/>
<b>hence </b><a NAME="E12:105_1_2_7_2_2_3"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><a NAME="E5:105_1_2_7_2"/><b>then </b><i><font color="Green">E638</font></i>: 
<font color="Maroon">L</font> is <a href="mesfunc5.html#V8" target="_self">convergent_to_+infty</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T13" target="_self">Th13</a></i>;<br/><b>hence </b><a NAME="E6:105_1_2_7_2"/>
<font color="Maroon">L</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E7:105_1_2_7_2"/><b>then </b>
(  ex <font color="Olive">g</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>(  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> &amp; ( for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> holds <br/> ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">p</font> ) &amp; <font color="Maroon">L</font> is <a href="mesfunc5.html#V7" target="_self">convergent_to_finite_number</a> ) or (  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">L</font> is <a href="mesfunc5.html#V8" target="_self">convergent_to_+infty</a> ) or (  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">L</font> is <a href="mesfunc5.html#V9" target="_self">convergent_to_-infty</a> ) )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T14" target="_self">Th14</a></i>;<br/><b>hence </b><a NAME="E8:105_1_2_7_2"/>
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E22:105_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="mesfunc5.html#D1" target="_self">Def1</a>, <a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
