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

<b>let </b><font color="Maroon">X</font> be    <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="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">S</font>, <font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">S</font>;<br/>







<b>assume </b><a NAME="E1:16"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &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 <br/> ( <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> ) ) )
 ;<br/>

<b>set </b><font color="Maroon">l</font> = <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E2:16"/><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> is    <a href="supinf_1.html#M2">minorant</a> of  <a href="supinf_2.html#K17">rng</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_1"/>
for <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>proof </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="E1:16_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<a NAME="E2:16_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  implies <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/>

<a NAME="E2:16_1_1_1_1"/><i><font color="Green">E30</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</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>then consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:16_1_1_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</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">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/>;<br/>
<a NAME="E6:16_1_1_1_1"/>
<font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="measure2.html#T1">MEASURE2:1</a></i>;<br/>
<a NAME="E7:16_1_1_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="supinf_2.html#T58">SUPINF_2:58</a></i>;<br/>
<a NAME="E8:16_1_1_1_1"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">x</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> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_1_1_1_1_1"/>
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">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0;<br/>
<a NAME="E1:16_1_1_1_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E2:16_1_1_1_1_1_1"/><i><font color="Green">E43</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="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_1_1_1_1"><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="E1:16_1_1_1_1_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0
 ;<br/>

<a NAME="E2:16_1_1_1_1_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:16_1_1_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="ref" href="measure3.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:16_1_1_1_1_1_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="measure3.html#E1:16"><i><font color="Green">E19</font></i></a>, );</i><br/>


</div><b>end;</b></div>
<a NAME="E2:16_1_1_1_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0
 
;<br/>
<a NAME="E3:16_1_1_1_1_1"/><b>then </b>
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span>
 
<b>by </b><i><a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/>
<a NAME="E4:16_1_1_1_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</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#T29">SUPINF_1:29</a></i>;<br/>
<i><font color="Green">E51</font></i>: <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="measure3.html#E1"><i><font color="Green">Lemma15</font></i></a>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="measure3.html#E1:16"><i><font color="Green">E19</font></i></a>, , <a class="ref" href="measure1.html#T63">MEASURE1:63</a></i>
<br/>.= 
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<b>set </b><font color="Maroon">k</font> = <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1;<br/>
<a NAME="E6:16_1_1_1_1_1"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="measure1.html#K1">.</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="measure3.html#E1"><i><font color="Green">Lemma15</font></i></a>, <a class="ref" href="measure3.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E7:16_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<b>hence </b><a NAME="E8:16_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">x</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> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="measure3.html#T3" target="_self">Th3</a>, <a class="ref" href="supinf_1.html#T76">SUPINF_1:76</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:16_1_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:16_1_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">x</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i>, , , <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a></i>;<br/>
<a NAME="E11:16_1_1_1_1"/><i><font color="Green">E55</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">c</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E13:16_1_1_1_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">c</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> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="measure3.html#E1:16"><i><font color="Green">E19</font></i></a>, </i>;<br/>
<a NAME="E14:16_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="measure3.html#T3" target="_self">Th3</a>, <a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="measure3.html#T3" target="_self">Th3</a>, <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>
<br/>.= 
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="measure3.html#T3" target="_self">Th3</a></i>
;<br/>
<a NAME="E16:16_1_1_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , , <a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a></i>;<br/>
<a NAME="E17:16_1_1_1_1"/><i><font color="Green">E60</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp;  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp;  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 
<b>by </b><i><a class="ref" href="measure3.html#T2" target="_self">Th2</a>, <a class="ref" href="supinf_1.html#T1">SUPINF_1:1</a>, <a class="ref" href="supinf_1.html#T6">SUPINF_1:6</a></i>;<br/>
<a NAME="E18:16_1_1_1_1"/>
<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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="measure3.html#T3" target="_self">Th3</a>, <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/>
<b>then </b><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> <span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x</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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="measure3.html#T3" target="_self">Th3</a>, <a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="measure3.html#T3" target="_self">Th3</a></i>
;<br/>
<b>hence </b><a NAME="E20:16_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:16_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:16_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:16_1"/>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> is    <a href="supinf_1.html#M2">minorant</a> of  <a href="supinf_2.html#K17">rng</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_1.html#D10">SUPINF_1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:16"/>
for <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">y</font> is    <a href="supinf_1.html#M2">minorant</a> of  <a href="supinf_2.html#K17">rng</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> holds <br/><font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">y</font> is    <a href="supinf_1.html#M2">minorant</a> of  <a href="supinf_2.html#K17">rng</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<a NAME="E2:16_2"/><i><font color="Green">E62</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K7">inf</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:16_2"/>
(  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> is   <a href="real_1.html#NM1">Real</a> &amp;  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> is   <a href="real_1.html#NM1">Real</a> )
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">c</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:16_2"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">a</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> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> )
 ;<br/>
<b>consider </b><font color="Maroon">s</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">t</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">r</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E7:16_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">s</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> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K7">inf</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> )
 ;<br/>
<a NAME="E8:16_2"/>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K7">inf</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="txt" href="measure3.html#E1"><i><font color="Green">Lemma15</font></i></a>, <a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/>
<b>then </b><i><font color="Green">E65</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><a href="supinf_1.html#K7">inf</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">r</font> = 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c</font>
<b>by </b><i><a class="txt" href="measure3.html#E1"><i><font color="Green">Lemma15</font></i></a>, , <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>
<br/>.= 
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="measure3.html#E1"><i><font color="Green">Lemma15</font></i></a></i>
;<br/>
<a NAME="E10:16_2"/>
<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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K7">inf</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="txt" href="measure3.html#E1"><i><font color="Green">Lemma15</font></i></a>, <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/>
<b>then </b><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> <span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="supinf_1.html#K7">inf</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K3">+</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="measure3.html#E1"><i><font color="Green">Lemma15</font></i></a>, <a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>
<br/>.= 
 <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="measure3.html#E1"><i><font color="Green">Lemma15</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E12:16_2"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="supinf_1.html#D17">SUPINF_1:def 17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:16"/>
 <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="supinf_1.html#D17">SUPINF_1:def 17</a></i>;<br/>


</div>
