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

<span class="kw">set </span><font color="Maroon" title="c3">G</font> = <font color="Maroon" title="c2">D</font>;<br/>
<a NAME="E1:31"/>
(  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">D</font></span>)</span> &amp; (  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c2">D</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">G</font> = <font color="Maroon" title="c2">D</font> as    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#D2" target="_self" title="MEASURE7:def.2">Def2</a></span>;<br/>
<a NAME="E3:31"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
 for <font color="Olive" title="b1">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b2">F</font> being    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Olive" title="b1">A</font> holds  <font color="Olive" title="b2">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a>  is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c5">A</font> holds  <font color="Olive" title="b1">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a>  is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c6">F</font> be    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c5">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a>  is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> </span><br/>



<a NAME="E1:31_1"/>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">n</font></span><br/>

<a NAME="E1:31_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K2" title="MEASURE5:func.2">diameter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="measure7.html#K2" title="MEASURE7:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:31_1_1"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#T13" title="MEASURE5:th.13">MEASURE5:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:31_1"/>
<font color="Maroon" title="c6">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a>  is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T39" title="SUPINF_2:th.39">SUPINF_2:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:31"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
 for <font color="Olive" title="b1">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> holds   <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">A</font></span><br/>

<a NAME="E1:31_2"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of  <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <font color="Maroon" title="c5">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be   <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xxreal_2.html#D2" title="XXREAL_2:def.2">XXREAL_2:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <font color="Maroon" title="c5">A</font> or  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <font color="Maroon" title="c5">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">x</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">F</font> being    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c5">A</font><span class="kw"> such that </span><br/><a NAME="E3:31_2_1"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure7.html#K6" title="MEASURE7:func.6">vol</a> <font color="Maroon" title="c7">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#D8" target="_self" title="MEASURE7:def.8">Def8</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">y</font> = <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>;<br/>
<a NAME="E4:31_2_1"/>
<font color="Maroon" title="c7">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a>  is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E28">A1</font></span></a></span>;<br/>
<a NAME="E5:31_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A5</font></span>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T40" title="SUPINF_2:th.40">SUPINF_2:40</a></span>;<br/>
<a NAME="E6:31_2_1"/>
(  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a>, <a class="ref" href="supinf_2.html#D17" title="SUPINF_2:def.17">SUPINF_2:def 17</a>, <a class="ref" href="xxreal_2.html#T4" title="XXREAL_2:th.4">XXREAL_2:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:31_2_1"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_2_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E5:31_2_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:31_2"/>
<a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <font color="Maroon" title="c5">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#D10" target="_self" title="MEASURE7:def.10">Def10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:31_2"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_2"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="xxreal_2.html#D4" title="XXREAL_2:def.4">XXREAL_2:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:31"/>
 <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a>  is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#D2" title="MEASURE1:def.2">MEASURE1:def 2</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="measure4.html#D1" title="MEASURE4:def.1">MEASURE4:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a>  is <a href="valued_0.html#V12" title="VALUED_0:attr.12">V74</a>() &amp; (  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive">b<sub>2</sub></font> or ( <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive">b<sub>2</sub></font> &amp; (  for <font color="Olive">b<sub>3</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) ) ) ) )</span><br/>

<a NAME="E6:31"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
 for <font color="Olive" title="b1">r</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )</span><br/>

<a NAME="E1:31_3"/>
 <a href="measure5.html#K2" title="MEASURE5:func.2">diameter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="measure7.html#K2" title="MEASURE7:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K2" title="MEASURE5:func.2">diameter</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:31_3"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a>, <a class="ref" href="measure5.html#T10" title="MEASURE5:th.10">MEASURE5:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span> <a href="measure7.html#K6" title="MEASURE7:func.6">vol</a> <font color="Maroon" title="c4">G</font> = 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="supinf_2.html#T48" title="SUPINF_2:th.48">SUPINF_2:48</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T44" title="SUPINF_2:th.44">SUPINF_2:44</a></span>
<br/>.= 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E30">A6</font></span></a></span>
;<br/>
<a NAME="E8:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A7</font></span>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#D8" target="_self" title="MEASURE7:def.8">Def8</a></span>;<br/>
<a NAME="E9:31"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of  <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be   <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xxreal_2.html#D2" title="XXREAL_2:def.2">XXREAL_2:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span> or  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_5"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font></span><br/>

<a NAME="E2:31_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A8</font></span>: 
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#T3" title="XXREAL_2:th.3">XXREAL_2:3</a></span>;<br/>
<a NAME="E3:31_5"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E29">A2</font></span></a></span>;<br/>
<a NAME="E4:31_5"/><span class="kw">then </span>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#D10" target="_self" title="MEASURE7:def.10">Def10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:31_5"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_5"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:31"/><span class="kw">then </span>
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:31"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="xxreal_2.html#T56" title="XXREAL_2:th.56">XXREAL_2:56</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:31"/>
<a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#D10" target="_self" title="MEASURE7:def.10">Def10</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="valued_0.html#D19" title="VALUED_0:def.19">VALUED_0:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive">b<sub>2</sub></font> or ( <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive">b<sub>2</sub></font> &amp; (  for <font color="Olive">b<sub>3</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font> or ( <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">B</font> &amp; (  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E12:31"/><span class="lab"><font color="Green" title="E32">A9</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">B</font> &amp; (  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )</span><br/>

<a NAME="E13:31"/><span class="lab"><font color="Green" title="E33">A10</font></span>: 
 <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <font color="Maroon" title="c5">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <font color="Maroon" title="c6">B</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <font color="Maroon" title="c5">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_6"/><span class="lab"><font color="Green" title="E34">A11</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <font color="Maroon" title="c6">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <font color="Maroon" title="c5">A</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">F</font> being    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c6">B</font><span class="kw"> such that </span><br/><a NAME="E4:31_6"/><span class="lab"><font color="Green" title="E35">A12</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure7.html#K6" title="MEASURE7:func.6">vol</a> <font color="Maroon" title="c9">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_6"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="ref" href="measure7.html#D8" target="_self" title="MEASURE7:def.8">Def8</a></span>;<br/>
<a NAME="E5:31_6"/>
<font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#D2" target="_self" title="MEASURE7:def.2">Def2</a></span>;<br/>
<a NAME="E6:31_6"/><span class="kw">then </span>
( (  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c9">F</font> <a href="measure7.html#K2" title="MEASURE7:func.2">.</a> <font color="Olive" title="b1">n</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a> ) &amp; <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">F</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:31"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">G</font> = <font color="Maroon" title="c9">F</font> as    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c5">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#D2" target="_self" title="MEASURE7:def.2">Def2</a></span>;<br/>
<a NAME="E8:31_6"/>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c11">n</font></span><br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c11">n</font> = 
 <a href="measure5.html#K2" title="MEASURE5:func.2">diameter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="measure7.html#K2" title="MEASURE7:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c11">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:31_6_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c11">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:31_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A13</font></span>: 
 <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T43" title="SUPINF_2:th.43">SUPINF_2:43</a></span>;<br/>
<a NAME="E10:31_6"/>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_6_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c11">n</font></span><br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c11">n</font> = 
 <a href="measure5.html#K2" title="MEASURE5:func.2">diameter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="measure7.html#K2" title="MEASURE7:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c11">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:31_6_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c11">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:31_6"/><span class="kw">then </span>
 <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">vol</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T43" title="SUPINF_2:th.43">SUPINF_2:43</a></span>;<br/>
<a NAME="E12:31_6"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure7.html#K6" title="MEASURE7:func.6">vol</a> <font color="Maroon" title="c10">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31_6"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="txt" href="#E9:31_6"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:31_6"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <font color="Maroon" title="c5">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#D8" target="_self" title="MEASURE7:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E14:31"/>
( <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <font color="Maroon" title="c5">A</font></span>)</span> &amp; <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <font color="Maroon" title="c6">B</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#D10" target="_self" title="MEASURE7:def.10">Def10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:31"/>
<a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:31"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="ref" href="xxreal_2.html#T60" title="XXREAL_2:th.60">XXREAL_2:60</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_7"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:31_7"/>
(  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds  <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st  <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_7_1"><span class="kw">suppose </span></a><a NAME="E1:31_7_1"/><span class="lab"><font color="Green" title="E34">A14</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds  <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span><br/><div class="add"><a NAME="E2:31_7_1"/><span class="lab"><font color="Green" title="E35">A15</font></span>: 
 <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#D17" title="SUPINF_2:def.17">SUPINF_2:def 17</a></span>;<br/><a NAME="E3:31_7_1"/><span class="lab"><font color="Green" title="E36">A16</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_7_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">n</font></span><br/>

<a NAME="E1:31_7_1_1"/>
<span class="p1">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:31_7_1_1"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E29">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:31_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A17</font></span>: 
<a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T39" title="SUPINF_2:th.39">SUPINF_2:39</a></span>;<br/><a NAME="E5:31_7_1"/>
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_7_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c8">y</font> =  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">x</font> =  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:31_7_1_2"/><span class="lab"><font color="Green" title="E38">A18</font></span>: 
 not  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:31_7_1_2"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_7_1"><span class="lab"><font color="Green" title="E36">A16</font></span></a></span>;<br/>
<a NAME="E3:31_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A19</font></span>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T44" title="SUPINF_2:th.44">SUPINF_2:44</a></span>;<br/>
<a NAME="E4:31_7_1_2"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a>, <a class="ref" href="xxreal_2.html#T4" title="XXREAL_2:th.4">XXREAL_2:4</a></span>;<br/>
<a NAME="E5:31_7_1_2"/><span class="kw">then </span>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_7_1_2"><span class="lab"><font color="Green" title="E39">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">eps</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E7:31_7_1_2"/><span class="lab"><font color="Green" title="E40">A20</font></span>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">eps</font>
 <span class="kw">and </span><br/><a NAME="E8:31_7_1_2"/><span class="lab"><font color="Green" title="E41">A21</font></span>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p4">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c10">eps</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_7_1_2"><span class="lab"><font color="Green" title="E38">A18</font></span></a>, <a class="ref" href="xxreal_3.html#T49" title="XXREAL_3:th.49">XXREAL_3:49</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">eps</font> = <font color="Maroon" title="c10">eps</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">E</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a><span class="kw"> such that </span><br/><a NAME="E11:31_7_1_2"/><span class="lab"><font color="Green" title="E42">A22</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">E</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font>
 <span class="kw">and </span><br/><a NAME="E12:31_7_1_2"/><span class="lab"><font color="Green" title="E43">A23</font></span>: 
 <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <font color="Maroon" title="c12">E</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">eps</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:31_7_1_2"><span class="lab"><font color="Green" title="E40">A20</font></span></a>, <a class="ref" href="measure6.html#T4" title="MEASURE6:th.4">MEASURE6:4</a></span>;<br/>
<a NAME="E13:31_7_1_2"/><span class="lab"><font color="Green" title="E44">A24</font></span>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p2">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <font color="Maroon" title="c12">E</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p2">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c11">eps</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:31_7_1_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="ref" href="xxreal_3.html#T36" title="XXREAL_3:th.36">XXREAL_3:36</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">F0</font> being    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1 st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F0</font> &amp;  <a href="measure7.html#K6" title="MEASURE7:func.6">vol</a> <font color="Olive" title="b1">F0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1</span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">E</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1</span>)</span> );<br/>
<a NAME="E14:31_7_1_2"/><span class="lab"><font color="Green" title="E45">A25</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">F0</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span>) st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">F0</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_7_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">F0</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span>) st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">n</font>,<font color="Olive" title="b1">F0</font>]</span><br/>

<a NAME="E1:31_7_1_2_1"/><span class="lab"><font color="Green" title="E46">A26</font></span>: 
( <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#D10" target="_self" title="MEASURE7:def.10">Def10</a></span>;<br/>
<a NAME="E2:31_7_1_2_1"/>
 <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_7_1"><span class="lab"><font color="Green" title="E34">A14</font></span></a></span>;<br/>
<a NAME="E3:31_7_1_2_1"/><span class="kw">then </span>
 not  <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<a NAME="E4:31_7_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:31_7_1_2_1"/><span class="lab"><font color="Green" title="E47">A27</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">x</font> = <font color="Maroon" title="c14">x</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31_7_1_2_1"><span class="lab"><font color="Green" title="E47">A27</font></span></a></span>;<br/>
<a NAME="E8:31_7_1_2_1"/>
 not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31_7_1_2_1"><span class="lab"><font color="Green" title="E47">A27</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E9:31_7_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A28</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E10:31_7_1_2_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31_7_1_2_1"><span class="lab"><font color="Green" title="E47">A27</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E11:31_7_1_2_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#T3" title="XXREAL_2:th.3">XXREAL_2:3</a></span>;<br/>
<a NAME="E12:31_7_1_2_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:31_7_1_2_1"><span class="lab"><font color="Green" title="E48">A28</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></span>;<br/>
<a NAME="E13:31_7_1_2_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E1:31_7_1_2_1"><span class="lab"><font color="Green" title="E46">A26</font></span></a>, <a class="ref" href="xxreal_0.html#T46" title="XXREAL_0:th.46">XXREAL_0:46</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">S</font> being   <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E15:31_7_1_2_1"/><span class="lab"><font color="Green" title="E49">A29</font></span>: 
<font color="Maroon" title="c16">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E16:31_7_1_2_1"/><span class="lab"><font color="Green" title="E50">A30</font></span>: 
<font color="Maroon" title="c16">S</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">E</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:31_7_1_2"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="ref" href="measure6.html#T5" title="MEASURE6:th.5">MEASURE6:5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">FS</font> being    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">n</font><span class="kw"> such that </span><br/><a NAME="E18:31_7_1_2_1"/><span class="lab"><font color="Green" title="E51">A31</font></span>: 
<font color="Maroon" title="c16">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure7.html#K6" title="MEASURE7:func.6">vol</a> <font color="Maroon" title="c17">FS</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:31_7_1_2_1"><span class="lab"><font color="Green" title="E49">A29</font></span></a>, <a class="ref" href="measure7.html#D8" target="_self" title="MEASURE7:def.8">Def8</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">FS</font> = <font color="Maroon" title="c17">FS</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span>) <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c18">FS</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">n</font>,<font color="Maroon" title="c18">FS</font>]</span><br/>

<span class="kw">thus </span><a NAME="E20:31_7_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">n</font>,<font color="Maroon" title="c18">FS</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:31_7_1_2_1"><span class="lab"><font color="Green" title="E50">A30</font></span></a>, <a class="txt" href="#E18:31_7_1_2_1"><span class="lab"><font color="Green" title="E51">A31</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c13">FF</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E16:31_7_1_2"/><span class="lab"><font color="Green" title="E46">A32</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E14:31_7_1_2"><span class="lab"><font color="Green" title="E45">A25</font></span></a>);<br/></span>
<a NAME="E17:31_7_1_2"/>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> is    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_7_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n</font> is    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n</font></span><br/>

<a NAME="E1:31_7_1_2_2"/>
 ex <font color="Olive" title="b1">F0</font> being    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n</font> st <br/>( <font color="Olive" title="b1">F0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n</font> &amp;  <a href="measure7.html#K6" title="MEASURE7:func.6">vol</a> <font color="Olive" title="b1">F0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">E</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">n</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:31_7_1_2"><span class="lab"><font color="Green" title="E46">A32</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:31_7_1_2_2"/>
<font color="Maroon" title="c13">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n</font> is    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">FF</font> = <font color="Maroon" title="c13">FF</font> as    <a href="measure7.html#M2" title="MEASURE7:mode.2">Interval_Covering</a> of <font color="Maroon" title="c7">F</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#D3" target="_self" title="MEASURE7:def.3">Def3</a></span>;<br/>
<a NAME="E19:31_7_1_2"/><span class="lab"><font color="Green" title="E47">A33</font></span>: 
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <font color="Maroon" title="c14">FF</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#T15" target="_self" title="MEASURE7:th.15">Th16</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">E</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1</span>)</span>;<br/>
<a NAME="E20:31_7_1_2"/><span class="lab"><font color="Green" title="E48">A34</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">y</font>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">G0</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a><span class="kw"> such that </span><br/><a NAME="E22:31_7_1_2"/><span class="lab"><font color="Green" title="E49">A35</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c15">G0</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E20:31_7_1_2"><span class="lab"><font color="Green" title="E48">A34</font></span></a>);<br/></span>
<a NAME="E23:31_7_1_2"/>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <font color="Maroon" title="c14">FF</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">G0</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_7_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <font color="Maroon" title="c14">FF</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">G0</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">n</font></span><br/>

<a NAME="E1:31_7_1_2_3"/>
(  ex <font color="Olive" title="b1">F0</font> being    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font> st <br/>( <font color="Olive" title="b1">F0</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c14">FF</font> <a href="measure7.html#K4" title="MEASURE7:func.4">.</a> <font color="Maroon" title="c16">n</font> &amp;  <a href="measure7.html#K6" title="MEASURE7:func.6">vol</a> <font color="Olive" title="b1">F0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">E</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">n</font></span>)</span> ) &amp; <span class="p1">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:31_7_1_2"><span class="lab"><font color="Green" title="E46">A32</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>;<br/>
<a NAME="E2:31_7_1_2_3"/><span class="kw">then </span>
 <a href="measure7.html#K6" title="MEASURE7:func.6">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">FF</font> <a href="measure7.html#K4" title="MEASURE7:func.4">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">E</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#D10" target="_self" title="MEASURE7:def.10">Def10</a></span>;<br/>
<a NAME="E3:31_7_1_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <font color="Maroon" title="c14">FF</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">E</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#D7" target="_self" title="MEASURE7:def.7">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:31_7_1_2_3"/>
<span class="p1">(<span class="default"><a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <font color="Maroon" title="c14">FF</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">G0</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c16">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:31_7_1_2"><span class="lab"><font color="Green" title="E49">A35</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E24:31_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A36</font></span>: 
 <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <font color="Maroon" title="c14">FF</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <font color="Maroon" title="c15">G0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T43" title="SUPINF_2:th.43">SUPINF_2:43</a></span>;<br/>
<a NAME="E25:31_7_1_2"/>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">E</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:31_7_1_2"><span class="lab"><font color="Green" title="E42">A22</font></span></a></span>;<br/>
<a NAME="E26:31_7_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">E</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T39" title="SUPINF_2:th.39">SUPINF_2:39</a></span>;<br/>
<a NAME="E27:31_7_1_2"/><span class="kw">then </span>
 <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <font color="Maroon" title="c15">G0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p2">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <font color="Maroon" title="c12">E</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31_7_1"><span class="lab"><font color="Green" title="E37">A17</font></span></a>, <a class="txt" href="#E22:31_7_1_2"><span class="lab"><font color="Green" title="E49">A35</font></span></a>, <a class="ref" href="measure7.html#T4" target="_self" title="MEASURE7:th.4">Th4</a></span>;<br/>
<a NAME="E28:31_7_1_2"/><span class="kw">then </span>
 <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <font color="Maroon" title="c15">G0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p2">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c11">eps</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:31_7_1_2"><span class="lab"><font color="Green" title="E44">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E29:31_7_1_2"/><span class="kw">then </span>
 <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <font color="Maroon" title="c14">FF</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p2">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c11">eps</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:31_7_1_2"><span class="lab"><font color="Green" title="E50">A36</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E30:31_7_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_7_1"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="txt" href="#E8:31_7_1_2"><span class="lab"><font color="Green" title="E41">A21</font></span></a>, <a class="txt" href="#E19:31_7_1_2"><span class="lab"><font color="Green" title="E47">A33</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:31_7_1"/>
<a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_7_1"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="ref" href="measure7.html#D10" target="_self" title="MEASURE7:def.10">Def10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_7_2"><span class="kw">suppose </span></a><a NAME="E1:31_7_2"/>
 ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st  <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:31_7_2"/><span class="lab"><font color="Green" title="E34">A37</font></span>: 
 <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><a NAME="E4:31_7_2"/>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_7_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">n</font></span><br/>

<a NAME="E1:31_7_2_1"/>
<span class="p1">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:31_7_2_1"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E29">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:31_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A38</font></span>: 
<a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T39" title="SUPINF_2:th.39">SUPINF_2:39</a></span>;<br/><a NAME="E6:31_7_2"/>
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#T13" title="XXREAL_2:th.13">XXREAL_2:13</a></span>;<br/><a NAME="E7:31_7_2"/><span class="kw">then </span>
<a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_7_2"><span class="lab"><font color="Green" title="E34">A37</font></span></a>, <a class="ref" href="measure7.html#D10" target="_self" title="MEASURE7:def.10">Def10</a></span>;<br/><a NAME="E8:31_7_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>;<br/><a NAME="E9:31_7_2"/><span class="kw">then </span>
 <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31_7_2"><span class="lab"><font color="Green" title="E35">A38</font></span></a>, <a class="ref" href="supinf_2.html#T45" title="SUPINF_2:th.45">SUPINF_2:45</a></span>;<br/><span class="kw">hence </span><a NAME="E10:31_7_2"/>
<a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K10" title="MEASURE7:func.10">OS_Meas</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
