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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="prob_1.html#NM1" title="PROB_1:NM.1">Field_Subset</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">M</font> being  <a href="measure1.html#NM1" title="MEASURE1:NM.1">Measure</a> of <font color="Olive" title="b1">F</font><br/> for <font color="Olive" title="b3">Sets</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Olive" title="b2">M</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</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" title="b3">Sets</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Olive" title="b2">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">Sets</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="prob_1.html#NM1" title="PROB_1:NM.1">Field_Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M</font> being  <a href="measure1.html#NM1" title="MEASURE1:NM.1">Measure</a> of <font color="Maroon" title="c2">F</font><br/> for <font color="Olive" title="b2">Sets</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Olive" title="b1">M</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</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" title="b2">Sets</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Olive" title="b1">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">Sets</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM1" title="MEASURE1:NM.1">Measure</a> of <font color="Maroon" title="c2">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Sets</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</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" title="b1">Sets</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">Sets</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">Sets</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</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="c4">Sets</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</font></span>)</span></span><br/>







<div><span class="lab"><font color="Green" title="E25">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:32_1"/><span class="lab"><font color="Green" title="E26">A2</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="measure8.html#K5" title="MEASURE8:func.5">Svc</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</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="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</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="c4">Sets</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</font></span>)</span></span><br/><a NAME="E2:32_1"/>
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> <font color="Maroon" title="c3">M</font>,<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="c4">Sets</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#K18" title="SUPINF_2:func.18">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</font></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c5">y</font> =  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> <font color="Maroon" title="c3">M</font>,<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="c4">Sets</font></span>)</span></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">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#K18" title="SUPINF_2:func.18">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E1:32_1_1"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</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#K18" title="SUPINF_2:func.18">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a>, <a class="ref" href="xxreal_2.html#T4" title="XXREAL_2:th.4">XXREAL_2:4</a></span>;<br/>
<a NAME="E2:32_1_1"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<span class="p1">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</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="measure8.html#T10" target="_self" title="MEASURE8:th.10">Th10</a>, <a class="ref" href="measure1.html#T54" title="MEASURE1:th.54">MEASURE1:54</a></span>;<br/>
<a NAME="E3:32_1_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.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="supinf_2.html#T58" title="SUPINF_2:th.58">SUPINF_2:58</a></span>;<br/>
<a NAME="E4:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A5</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">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#K19" title="SUPINF_2:func.19">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.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="supinf_2.html#T63" title="SUPINF_2:th.63">SUPINF_2:63</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:32_1_1"/>
not  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> <font color="Maroon" title="c3">M</font>,<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="c4">Sets</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#K18" title="SUPINF_2:func.18">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</font></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">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="E7:32_1_1"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">eps</font>
 <span class="kw">and </span><br/><a NAME="E8:32_1_1"/><span class="lab"><font color="Green" title="E31">A7</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#K18" title="SUPINF_2:func.18">rng</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</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="c7">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="measure8.html#K5" title="MEASURE8:func.5">Svc</a> <font color="Maroon" title="c3">M</font>,<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="c4">Sets</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E1:32_1_1"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="xxreal_3.html#T55" title="XXREAL_3:th.55">XXREAL_3:55</a></span>;<br/>
<a NAME="E9:32_1_1"/>
<font color="Maroon" title="c7">eps</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">then </span><span class="kw">consider </span><font color="Maroon" title="c8">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="E11:32_1_1"/><span class="lab"><font color="Green" title="E32">A8</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="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">E</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font>
 <span class="kw">and </span><br/><a NAME="E12:32_1_1"/><span class="lab"><font color="Green" title="E33">A9</font></span>: 
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <font color="Maroon" title="c8">E</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">eps</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="measure6.html#T31" title="MEASURE6:th.31">MEASURE6:31</a></span>;<br/>
<a NAME="E13:32_1_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="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">E</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:32_1_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a></span>;<br/>
<a NAME="E14:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A10</font></span>: 
<font color="Maroon" title="c8">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#T58" title="SUPINF_2:th.58">SUPINF_2:58</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="measure8.html#M3" title="MEASURE8:mode.3">Covering</a> of <font color="Maroon" title="c4">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> $1,<font color="Maroon" title="c2">F</font> st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F0</font> &amp;  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">F0</font></span>)</span> <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="measure8.html#K5" title="MEASURE8:func.5">Svc</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> $1</span>)</span></span>)</span></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> $1</span>)</span> );<br/>
<a NAME="E15:32_1_1"/><span class="lab"><font color="Green" title="E35">A11</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> <font color="Maroon" title="c1">X</font></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="32_1_1_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="Red">::  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> <font color="Maroon" title="c1">X</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">n</font>,<font color="Olive" title="b1">F0</font>]</span><br/>

<a NAME="E1:32_1_1_1"/>
(  <a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  &amp; <span class="p1">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c9">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="measure8.html#K5" title="MEASURE8:func.5">Svc</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D8" target="_self" title="MEASURE8:def.8">Def8</a>, <a class="ref" href="measure8.html#T10" target="_self" title="MEASURE8:th.10">Th10</a></span>;<br/>
<a NAME="E2:32_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A12</font></span>: 
(  <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> &amp;  <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="measure8.html#K5" title="MEASURE8:func.5">Svc</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T70" title="SUPINF_2:th.70">SUPINF_2:70</a></span>;<br/>
<a NAME="E3:32_1_1_1"/>
 <a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E26">A2</font></span></a></span>;<br/>
<a NAME="E4:32_1_1_1"/><span class="kw">then </span>
not  <a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></span>;<br/>
<a NAME="E5:32_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</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="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:32_1_1_1"/><span class="lab"><font color="Green" title="E37">A13</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</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="c11">x</font> = <font color="Maroon" title="c10">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="#E7:32_1_1_1"><span class="lab"><font color="Green" title="E37">A13</font></span></a></span>;<br/>
<a NAME="E9:32_1_1_1"/>
not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_1_1"><span class="lab"><font color="Green" title="E37">A13</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="E10:32_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A14</font></span>: 
<font color="Maroon" title="c11">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="E11:32_1_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_1_1"><span class="lab"><font color="Green" title="E37">A13</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="E12:32_1_1_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="measure8.html#K5" title="MEASURE8:func.5">Svc</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">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="E13:32_1_1_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="measure8.html#K5" title="MEASURE8:func.5">Svc</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c9">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="#E10:32_1_1_1"><span class="lab"><font color="Green" title="E38">A14</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="E14:32_1_1_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="measure8.html#K5" title="MEASURE8:func.5">Svc</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c9">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="#E2:32_1_1_1"><span class="lab"><font color="Green" title="E36">A12</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="c12">S1</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="E16:32_1_1_1"/><span class="lab"><font color="Green" title="E39">A15</font></span>: 
<font color="Maroon" title="c12">S1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E17:32_1_1_1"/><span class="lab"><font color="Green" title="E40">A16</font></span>: 
<font color="Maroon" title="c12">S1</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="measure8.html#K5" title="MEASURE8:func.5">Svc</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:32_1_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="ref" href="measure6.html#T32" title="MEASURE6:th.32">MEASURE6:32</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">FS</font> being    <a href="measure8.html#M3" title="MEASURE8:mode.3">Covering</a> of <font color="Maroon" title="c4">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c2">F</font><span class="kw"> such that </span><br/><a NAME="E19:32_1_1_1"/><span class="lab"><font color="Green" title="E41">A17</font></span>: 
<font color="Maroon" title="c12">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c13">FS</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:32_1_1_1"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="ref" href="measure8.html#D7" target="_self" title="MEASURE8:def.7">Def7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">FS</font> = <font color="Maroon" title="c13">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> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c14">FS</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c14">FS</font>]</span><br/>

<span class="kw">thus </span><a NAME="E21:32_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c14">FS</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:32_1_1_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E19:32_1_1_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c9">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> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E17:32_1_1"/><span class="lab"><font color="Green" title="E36">A18</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="c9">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="#E15:32_1_1"><span class="lab"><font color="Green" title="E35">A11</font></span></a>);<br/></span>
<a NAME="E18:32_1_1"/><span class="lab"><font color="Green" title="E37">A19</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c9">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> is    <a href="measure8.html#M3" title="MEASURE8:mode.3">Covering</a> of <font color="Maroon" title="c4">Sets</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font> is    <a href="measure8.html#M3" title="MEASURE8:mode.3">Covering</a> of <font color="Maroon" title="c4">Sets</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c2">F</font></span><br/>

<a NAME="E1:32_1_1_2"/>
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E2:32_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">F0</font> being   <a href="measure8.html#M3" title="MEASURE8:mode.3">Covering</a> of <font color="Maroon" title="c4">Sets</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c2">F</font> st <br/>( <font color="Olive" title="b1">F0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font> &amp;  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">F0</font></span>)</span> <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="measure8.html#K5" title="MEASURE8:func.5">Svc</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">Sets</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:32_1_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:32_1_1_2"/>
<font color="Maroon" title="c9">FF</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font> is    <a href="measure8.html#M3" title="MEASURE8:mode.3">Covering</a> of <font color="Maroon" title="c4">Sets</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">deffunc </span><font color="Maroon">H<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> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> $1</span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> $1</span>)</span>;<br/>
<a NAME="E19:32_1_1"/><span class="lab"><font color="Green" title="E38">A20</font></span>: 
for <font color="Olive" title="b1">x</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">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) is    <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="c10">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="E21:32_1_1"/><span class="lab"><font color="Green" title="E39">A21</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" title="c10">G0</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S9" title="FUNCT_2:sch.9">FUNCT_2:sch 9</a>(<a class="txt" href="#E19:32_1_1"><span class="lab"><font color="Green" title="E38">A20</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c11">FF</font> = <font color="Maroon" title="c9">FF</font> as    <a href="measure8.html#M4" title="MEASURE8:mode.4">Covering</a> of <font color="Maroon" title="c4">Sets</font>,<font color="Maroon" title="c2">F</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:32_1_1"><span class="lab"><font color="Green" title="E37">A19</font></span></a>, <a class="ref" href="measure8.html#D4" target="_self" title="MEASURE8:def.4">Def4</a></span>;<br/>
<a NAME="E23:32_1_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  <span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c11">FF</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</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="c10">G0</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c11">FF</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">G0</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font></span><br/>

<a NAME="E1:32_1_1_3"/>
(  ex <font color="Olive" title="b1">F0</font> being   <a href="measure8.html#M3" title="MEASURE8:mode.3">Covering</a> of <font color="Maroon" title="c4">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c2">F</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="c11">FF</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c12">n</font> &amp;  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">F0</font></span>)</span> <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="measure8.html#K5" title="MEASURE8:func.5">Svc</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> ) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:32_1_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>;<br/>
<a NAME="E2:32_1_1_3"/><span class="kw">then </span>
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">FF</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c12">n</font></span>)</span></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"><span class="p3">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D8" target="_self" title="MEASURE8:def.8">Def8</a></span>;<br/>
<a NAME="E3:32_1_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c11">FF</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">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"><span class="p3">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D6" target="_self" title="MEASURE8:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:32_1_1_3"/>
<span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c11">FF</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">G0</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:32_1_1"><span class="lab"><font color="Green" title="E39">A21</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E24:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A22</font></span>: 
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c11">FF</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <font color="Maroon" title="c10">G0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T62" title="SUPINF_2:th.62">SUPINF_2:62</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E41">A23</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">G0</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span><br/><a NAME="E1:32_1_1_4"/>
<font color="Maroon" title="c12">n</font> is    <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">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E2:32_1_1_4"/>
<font color="Maroon" title="c10">G0</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:32_1_1"><span class="lab"><font color="Green" title="E39">A21</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E26:32_1_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</font></span>)</span></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <font color="Maroon" title="c8">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#K20" title="SUPINF_2:func.20">SUM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</font></span>)</span></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c7">eps</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32_1_1"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="ref" href="xxreal_3.html#T38" title="XXREAL_3:th.38">XXREAL_3:38</a></span>;<br/>
<a NAME="E27:32_1_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <font color="Maroon" title="c10">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#K20" title="SUPINF_2:func.20">SUM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</font></span>)</span></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c7">eps</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E14:32_1_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E25:32_1_1"><span class="lab"><font color="Green" title="E41">A23</font></span></a>, <a class="ref" href="measure8.html#T3" target="_self" title="MEASURE8:th.3">Th3</a></span>;<br/>
<a NAME="E28:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A24</font></span>: 
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c11">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#K20" title="SUPINF_2:func.20">SUM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</font></span>)</span></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c7">eps</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:32_1_1"><span class="lab"><font color="Green" title="E40">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E29:32_1_1"/>
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> <font color="Maroon" title="c3">M</font>,<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="c4">Sets</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#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c11">FF</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T7" target="_self" title="MEASURE8:th.7">Th7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E30:32_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E28:32_1_1"><span class="lab"><font color="Green" title="E42">A24</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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:32_1"/>
<span class="p1">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</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="c4">Sets</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D8" target="_self" title="MEASURE8:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:32_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="measure8.html#K5" title="MEASURE8:func.5">Svc</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</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="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</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="c4">Sets</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">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="E3:32_2"/><span class="lab"><font color="Green" title="E26">A25</font></span>: 
 <a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><a NAME="E4:32_2"/>
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</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="E5:32_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">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="#E3:32_2"><span class="lab"><font color="Green" title="E26">A25</font></span></a>, <a class="ref" href="measure8.html#D8" target="_self" title="MEASURE8:def.8">Def8</a></span>;<br/><span class="kw">hence </span><a NAME="E6:32_2"/>
<span class="p1">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</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="c4">Sets</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="measure8.html#E24"><span class="lab"><font color="Green" title="E23">Lm2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:32"/>
<span class="p1">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</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="c4">Sets</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K6" title="MEASURE8:func.6">C_Meas</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">Sets</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E25">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
