<?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">C</font> being   <a href="measure4.html#M1" title="MEASURE4:mode.1">C_Measure</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">seq</font> being  <a href="measure8.html#NM3" title="MEASURE8:NM.3">Sep_Sequence</a> of <span class="p1">(<span class="default"><a href="measure4.html#K1" title="MEASURE4:func.1">sigma_Field</a> <font color="Olive" title="b1">C</font></span>)</span> holds <br/> (  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Olive" title="b2">seq</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure4.html#K1" title="MEASURE4:func.1">sigma_Field</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b1">C</font> <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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Olive" title="b2">seq</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc9.html#K3" title="MESFUNC9:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">seq</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">C</font> be    <a href="measure4.html#M1" title="MEASURE4:mode.1">C_Measure</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">seq</font> being  <a href="measure8.html#NM3" title="MEASURE8:NM.3">Sep_Sequence</a> of <span class="p1">(<span class="default"><a href="measure4.html#K1" title="MEASURE4:func.1">sigma_Field</a> <font color="Maroon" title="c2">C</font></span>)</span> holds <br/> (  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Olive" title="b1">seq</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure4.html#K1" title="MEASURE4:func.1">sigma_Field</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Maroon" title="c2">C</font> <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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Olive" title="b1">seq</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc9.html#K3" title="MESFUNC9:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">seq</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">seq</font> be   <a href="measure8.html#NM3" title="MEASURE8:NM.3">Sep_Sequence</a> of <span class="p1">(<span class="default"><a href="measure4.html#K1" title="MEASURE4:func.1">sigma_Field</a> <font color="Maroon" title="c2">C</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure4.html#K1" title="MEASURE4:func.1">sigma_Field</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Maroon" title="c2">C</font> <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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc9.html#K3" title="MESFUNC9:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> )</span><br/>





<a NAME="E1:47"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
 <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="measure4.html#K1" title="MEASURE4:func.1">sigma_Field</a> <font color="Maroon" title="c2">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">seq1</font> = <font color="Maroon" title="c3">seq</font> as   <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> <font color="Maroon" title="c1">X</font></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/>
<a NAME="E3:47"/><span class="lab"><font color="Green" title="E38">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 <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">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"><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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<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 <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">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"><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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">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"><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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">A</font></span><br/>



<a NAME="E1:47_1"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
for <font color="Olive" title="b1">k</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 <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">k</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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_1_1"/><span class="lab"><font color="Green" title="E40">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</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="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<div><span class="lab"><font color="Green" title="E41">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">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"> <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c7">k</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><br/><a NAME="E1:47_1_1_1"/><span class="lab"><font color="Green" title="E42">A6</font></span>: 
<font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c7">k</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a></span>;<br/><a NAME="E2:47_1_1_1"/>
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">m</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">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c9">m</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">m1</font> = <font color="Maroon" title="c9">m</font> as    <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">assume </span><a NAME="E2:47_1_1_1_1"/>
<font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c9">m</font></span><br/>

<a NAME="E3:47_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E4:47_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">m</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_2.html#D3" title="PROB_2:def.3">PROB_2:def 3</a></span>;<br/>
<a NAME="E5:47_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a>, <a class="ref" href="xboole_1.html#T86" title="XBOOLE_1:th.86">XBOOLE_1:86</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c10">m1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_2.html#D5" title="SETLIM_2:def.5">SETLIM_2:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>
;<br/>
<a NAME="E7:47_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c10">m1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47_1_1_1_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="ref" href="setlim_2.html#D5" title="SETLIM_2:def.5">SETLIM_2:def 5</a></span>;<br/>
<a NAME="E8:47_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c10">m1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:47_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c9">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</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="E3:47_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A8</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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c7">k</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">Ser</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c7">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T23" target="_self" title="MEASURE8:th.23">Th23</a></span>;<br/><a NAME="E4:47_1_1_1"/>
<font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></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="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/><a NAME="E5:47_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a>, <a class="ref" href="xboole_1.html#T34" title="XBOOLE_1:th.34">XBOOLE_1:34</a></span>;<br/><a NAME="E6:47_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>;<br/><a NAME="E7:47_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c7">k</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c7">k</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_1_1"><span class="lab"><font color="Green" title="E43">A8</font></span></a>, <a class="ref" href="xxreal_3.html#T38" title="XXREAL_3:th.38">XXREAL_3:38</a></span>;<br/><span class="kw">hence </span><a NAME="E8:47_1_1_1"/>
<font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c7">k</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1_1"><span class="lab"><font color="Green" title="E42">A6</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">let </span><font color="Maroon" title="c8">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">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"> <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><br/>

<font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T49" title="XBOOLE_1:th.49">XBOOLE_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c8">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<a NAME="E4:47_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A9</font></span>: 
<font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c7">k</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1_1"><span class="lab"><font color="Green" title="E41">A5</font></span></a></span>;<br/>
<a NAME="E5:47_1_1"/><span class="lab"><font color="Green" title="E43">A10</font></span>: 
<font color="Maroon" title="c8">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T33" title="XBOOLE_1:th.33">XBOOLE_1:33</a></span>;<br/>
<a NAME="E6:47_1_1"/><span class="lab"><font color="Green" title="E44">A11</font></span>: 
<font color="Maroon" title="c8">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E7:47_1_1"/>
( <font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font> &amp; <font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> = 
<font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E5:47_1_1"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="ref" href="measure4.html#T14" title="MEASURE4:th.14">MEASURE4:14</a></span>
<br/>.= 
<font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T24" title="XBOOLE_1:th.24">XBOOLE_1:24</a></span>
<br/>.= 
<font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">A</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></span>
;<br/>
<a NAME="E9:47_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:47_1_1"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E10:47_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A12</font></span>: 
<font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p0">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_1_1"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="ref" href="xxreal_3.html#T38" title="XXREAL_3:th.38">XXREAL_3:38</a></span>;<br/>
<span class="lab"><font color="Green" title="E46">A13</font></span>: <span class="p1">(<span class="default"><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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c7">k</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c7">k</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_2.html#D5" title="SETLIM_2:def.5">SETLIM_2:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<br/>.= 
<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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<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/>
<a NAME="E12:47_1_1"/><span class="lab"><font color="Green" title="E47">A14</font></span>: 
<font color="Maroon" title="c2">C</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="measure4.html#D2" title="MEASURE4:def.2">MEASURE4:def 2</a></span>;<br/>
<a NAME="E13:47_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A15</font></span>: 
<font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span> 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#T54" title="MEASURE1:th.54">MEASURE1:54</a></span>;<br/>
<a NAME="E14:47_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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#T70" title="SUPINF_2:th.70">SUPINF_2:70</a></span>;<br/>
<a NAME="E15:47_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A16</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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:47_1_1"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="ref" href="measure7.html#T2" title="MEASURE7:th.2">MEASURE7:2</a></span>;<br/>
<a NAME="E16:47_1_1"/>
( <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:47_1_1"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="ref" href="supinf_2.html#T70" title="SUPINF_2:th.70">SUPINF_2:70</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:47_1_1"/>
<font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:47_1_1"><span class="lab"><font color="Green" title="E49">A16</font></span></a>, <a class="txt" href="#E10:47_1_1"><span class="lab"><font color="Green" title="E45">A12</font></span></a>, <a class="txt" href="#E11:47_1_1"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="ref" href="xxreal_3.html#T30" title="XXREAL_3:th.30">XXREAL_3:30</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="E2:47_1"/><span class="lab"><font color="Green" title="E40">A17</font></span>: 
<font color="Maroon" title="c3">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure4.html#K1" title="MEASURE4:func.1">sigma_Field</a> <font color="Maroon" title="c2">C</font>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">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"> <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c7">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></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>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><br/><a NAME="E1:47_1_2"/>
( <font color="Maroon" title="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A18</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> = 
<font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="ref" href="measure4.html#T14" title="MEASURE4:th.14">MEASURE4:14</a></span>
<br/>.= 
<font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T49" title="XBOOLE_1:th.49">XBOOLE_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
<br/>.= 
<font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c7">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T51" title="XBOOLE_1:th.51">XBOOLE_1:51</a></span>
;<br/><a NAME="E3:47_1_2"/>
<font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_1.html#T1" title="ABCMIZ_1:th.1">ABCMIZ_1:1</a></span>;<br/><a NAME="E4:47_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></span>;<br/><a NAME="E5:47_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T34" title="XBOOLE_1:th.34">XBOOLE_1:34</a></span>;<br/><a NAME="E6:47_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></span>;<br/><a NAME="E7:47_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A19</font></span>: 
<font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure4.html#D2" title="MEASURE4:def.2">MEASURE4:def 2</a></span>;<br/><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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></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="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</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/>.= 
<font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<br/>.= 
<font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_2.html#D5" title="SETLIM_2:def.5">SETLIM_2:def 5</a></span>
;<br/><span class="kw">hence </span><a NAME="E9:47_1_2"/>
<font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c7">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></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>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1_2"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E7:47_1_2"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="ref" href="xxreal_3.html#T38" title="XXREAL_3:th.38">XXREAL_3:38</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="E4:47_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A20</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E5:47_1"/>
for <font color="Olive" title="b1">k</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">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E4:47_1"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="txt" href="#E1:47_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:47_1"/>
<span class="p1">(<span class="default"><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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">A</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>
<a NAME="E4:47"/><span class="lab"><font color="Green" title="E39">A21</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 <font color="Maroon" title="c1">X</font> holds <br/> ( <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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_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 <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="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> )</span><br/>

<a NAME="E1:47_2"/><span class="lab"><font color="Green" title="E40">A22</font></span>: 
<font color="Maroon" title="c2">C</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="measure4.html#D2" title="MEASURE4:def.2">MEASURE4:def 2</a></span>;<br/>
<a NAME="E2:47_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A23</font></span>: 
<font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span> 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#T54" title="MEASURE1:th.54">MEASURE1:54</a></span>;<br/>
<a NAME="E3:47_2"/><span class="lab"><font color="Green" title="E42">A24</font></span>: 
<font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_2"><span class="lab"><font color="Green" title="E40">A22</font></span></a>, <a class="ref" href="supinf_2.html#T70" title="SUPINF_2:th.70">SUPINF_2:70</a></span>;<br/>
<a NAME="E4:47_2"/>
( (  not <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or  not <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></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>  ) implies <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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:47_2_1"/><span class="lab"><font color="Green" title="E43">A25</font></span>: 
(  not <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or  not <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></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>  )
 ; <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="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">A</font></span><br/>

<a NAME="E2:47_2_1"/>
for <font color="Olive" title="b1">x</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>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_1_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> ; <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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> implies <font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_2_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</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"> <font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:47_2_1_1"/><span class="lab"><font color="Green" title="E44">A26</font></span>: 
<font color="Maroon" title="c7">i</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">and </span><br/><a NAME="E4:47_2_1_1"/><span class="lab"><font color="Green" title="E45">A27</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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T17" title="FUNCT_2:th.17">FUNCT_2:17</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">i</font> = <font color="Maroon" title="c7">i</font> as    <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="txt" href="#E3:47_2_1_1"><span class="lab"><font color="Green" title="E44">A26</font></span></a></span>;<br/>
<a NAME="E6:47_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_2"><span class="lab"><font color="Green" title="E41">A23</font></span></a>, <a class="ref" href="supinf_2.html#T70" title="SUPINF_2:th.70">SUPINF_2:70</a></span>;<br/>
<a NAME="E7:47_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A28</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_2"><span class="lab"><font color="Green" title="E41">A23</font></span></a>, <a class="txt" href="#E4:47_2_1_1"><span class="lab"><font color="Green" title="E45">A27</font></span></a>, <a class="ref" href="measure7.html#T2" title="MEASURE7:th.2">MEASURE7:2</a></span>;<br/>
<a NAME="E8:47_2_1_1"/>
( <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <span class="p1">(<span class="default"><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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">i</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="txt" href="#E1:47_2"><span class="lab"><font color="Green" title="E40">A22</font></span></a>, <a class="ref" href="supinf_2.html#T70" title="SUPINF_2:th.70">SUPINF_2:70</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:47_2_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_2_1"><span class="lab"><font color="Green" title="E43">A25</font></span></a>, <a class="txt" href="#E4:47_2_1_1"><span class="lab"><font color="Green" title="E45">A27</font></span></a>, <a class="txt" href="#E7:47_2_1_1"><span class="lab"><font color="Green" title="E46">A28</font></span></a>, <a class="ref" href="xxreal_3.html#T67" title="XXREAL_3:th.67">XXREAL_3:67</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="E3:47_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> is    <a href="xxreal_2.html#M1" title="XXREAL_2:mode.1">UpperBound</a> of  <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D1" title="XXREAL_2:def.1">XXREAL_2:def 1</a></span>;<br/>
<a NAME="E4:47_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A29</font></span>: 
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D3" title="XXREAL_2:def.3">XXREAL_2:def 3</a></span>;<br/>
<a NAME="E5:47_2_1"/>
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_2"><span class="lab"><font color="Green" title="E41">A23</font></span></a>, <a class="ref" href="measure6.html#T2" title="MEASURE6:th.2">MEASURE6:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:47_2_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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_2"><span class="lab"><font color="Green" title="E42">A24</font></span></a>, <a class="txt" href="#E4:47_2_1"><span class="lab"><font color="Green" title="E44">A29</font></span></a>, <a class="ref" href="xxreal_3.html#T66" title="XXREAL_3:th.66">XXREAL_3:66</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="E5:47_2"/>
<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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T3" title="XXREAL_0:th.3">XXREAL_0:3</a></span>; <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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span><br/>

<a NAME="E6:47_2"/>
<font color="Maroon" title="c2">C</font> <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> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure4.html#D2" title="MEASURE4:def.2">MEASURE4:def 2</a></span>;<br/>
<a NAME="E7:47_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:47_2"/>
<font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_2.html#T38" title="SETLIM_2:th.38">SETLIM_2:38</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="E5:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A30</font></span>: 
<font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E6:47"/>
for <font color="Olive" title="b1">W</font>, <font color="Olive" title="b2">Z</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font> &amp; <font color="Olive" title="b2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">W</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">Z</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">W</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">Z</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">W</font>, <font color="Maroon" title="c6">Z</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">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"> ( <font color="Maroon" title="c5">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font> &amp; <font color="Maroon" title="c6">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">W</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">Z</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">Z</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:47_3"/><span class="lab"><font color="Green" title="E41">A31</font></span>: 
<font color="Maroon" title="c5">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font>
 <span class="kw">and </span><br/><a NAME="E2:47_3"/><span class="lab"><font color="Green" title="E42">A32</font></span>: 
<font color="Maroon" title="c6">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font></span>)</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"><font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">W</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">Z</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">Z</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c5">W</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">Z</font>;<br/>
<a NAME="E3:47_3"/><span class="lab"><font color="Green" title="E43">A33</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">Z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Z</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_3"><span class="lab"><font color="Green" title="E42">A32</font></span></a>, <a class="ref" href="xboole_1.html#T30" title="XBOOLE_1:th.30">XBOOLE_1:30</a></span>;<br/>
<a NAME="E4:47_3"/>
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/>
<a NAME="E5:47_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A34</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E6:47_3"/>
<font color="Maroon" title="c5">W</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_3"><span class="lab"><font color="Green" title="E41">A31</font></span></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></span>;<br/>
<a NAME="E7:47_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">W</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47_3"><span class="lab"><font color="Green" title="E44">A34</font></span></a></span>;<br/>
<a NAME="E8:47_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A35</font></span>: 
<font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">Z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_3"><span class="lab"><font color="Green" title="E43">A33</font></span></a>, <a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></span>;<br/>
<a NAME="E9:47_3"/>
<font color="Maroon" title="c6">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_3"><span class="lab"><font color="Green" title="E42">A32</font></span></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></span>;<br/>
<a NAME="E10:47_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A36</font></span>: 
<font color="Maroon" title="c6">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47_3"><span class="lab"><font color="Green" title="E44">A34</font></span></a></span>;<br/>
<a NAME="E11:47_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">Z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">W</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_3"><span class="lab"><font color="Green" title="E41">A31</font></span></a>, <a class="ref" href="xboole_1.html#T30" title="XBOOLE_1:th.30">XBOOLE_1:30</a></span>;<br/>
<a NAME="E12:47_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">W</font> <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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">Z</font></span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E39">A21</font></span></a>, <a class="txt" href="#E10:47_3"><span class="lab"><font color="Green" title="E46">A36</font></span></a></span>;<br/>
<a NAME="E13:47_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A37</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">W</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">Z</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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">Z</font></span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">Z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:47_3"><span class="lab"><font color="Green" title="E45">A35</font></span></a>, <a class="ref" href="xxreal_3.html#T38" title="XXREAL_3:th.38">XXREAL_3:38</a></span>;<br/>
<a NAME="E14:47_3"/>
<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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">Z</font></span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">Z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">Z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E39">A21</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:47_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">W</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">Z</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">Z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:47_3"><span class="lab"><font color="Green" title="E47">A37</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>
<a NAME="E7:47"/><span class="kw">then </span>
 <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure4.html#K1" title="MEASURE4:func.1">sigma_Field</a> <font color="Maroon" title="c2">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure4.html#D3" title="MEASURE4:def.3">MEASURE4:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:47"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure4.html#K1" title="MEASURE4:func.1">sigma_Field</a> <font color="Maroon" title="c2">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></span>; <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="c2">C</font> <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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc9.html#K3" title="MESFUNC9:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">seq</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">Sseq</font> =  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">seq1</font></span>)</span>;<br/>
<a NAME="E9:47"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/>
<a NAME="E10:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A38</font></span>: 
<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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E11:47"/>
<font color="Maroon" title="c2">C</font> is  <a href="valued_0.html#V12" title="VALUED_0:attr.12">zeroed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure4.html#D2" title="MEASURE4:def.2">MEASURE4:def 2</a></span>;<br/>
<a NAME="E12:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A39</font></span>: 
<font color="Maroon" title="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span> <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="txt" href="#E10:47"><span class="lab"><font color="Green" title="E41">A38</font></span></a>, <a class="ref" href="valued_0.html#D19" title="VALUED_0:def.19">VALUED_0:def 19</a></span>;<br/>
<a NAME="E13:47"/>
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><span class="p1">(<span class="default"><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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">n</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"> ( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies <span class="p1">(<span class="default"><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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_4"/>
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">n1</font> = <font color="Maroon" title="c6">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E3:47_4"/>
<font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c7">n1</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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></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="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/>
<a NAME="E4:47_4"/><span class="kw">then </span>
<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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c7">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:47_4"/>
<span class="p1">(<span class="default"><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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_2.html#D5" title="SETLIM_2:def.5">SETLIM_2:def 5</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="E14:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A40</font></span>: 
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K1" title="RINFSUP2:func.1">sup</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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></span>;<br/>
<a NAME="E15:47"/>
<font color="Maroon" title="c2">C</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="measure4.html#D2" title="MEASURE4:def.2">MEASURE4:def 2</a></span>;<br/>
<a NAME="E16:47"/><span class="kw">then </span>
<font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">seq1</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="measure1.html#T54" title="MEASURE1:th.54">MEASURE1:54</a></span>;<br/>
<a NAME="E17:47"/><span class="kw">then </span>
for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</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>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> holds <br/><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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">m</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">Ser</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#T9" title="MEASURE7:th.9">MEASURE7:9</a></span>;<br/>
<a NAME="E18:47"/><span class="kw">then </span>
 <a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">seq1</font></span>)</span> is  <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D15" title="VALUED_0:def.15">VALUED_0:def 15</a></span>;<br/>
<a NAME="E19:47"/><span class="kw">then </span>
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:47"><span class="lab"><font color="Green" title="E43">A40</font></span></a>, <a class="ref" href="rinfsup2.html#T37" title="RINFSUP2:th.37">RINFSUP2:37</a></span>;<br/>
<a NAME="E20:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A41</font></span>: 
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">seq1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T1" target="_self" title="MEASURE8:th.1">Th1</a></span>;<br/>
<a NAME="E21:47"/>
<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"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c4">seq1</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="c2">C</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p5">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">C</font> <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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E39">A21</font></span></a></span>;<br/>
<a NAME="E22:47"/><span class="kw">then </span>
(  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">seq1</font> &amp; <font color="Maroon" title="c2">C</font> <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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="mesfunc9.html#K3" title="MESFUNC9:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:47"><span class="lab"><font color="Green" title="E42">A39</font></span></a>, <a class="txt" href="#E20:47"><span class="lab"><font color="Green" title="E44">A41</font></span></a>, <a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a>, <a class="ref" href="xxreal_3.html#T4" title="XXREAL_3:th.4">XXREAL_3:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:47"/>
<font color="Maroon" title="c2">C</font> <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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc9.html#K3" title="MESFUNC9:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">seq</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47"><span class="lab"><font color="Green" title="E40">A30</font></span></a>, <a class="txt" href="#E20:47"><span class="lab"><font color="Green" title="E44">A41</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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>
