<?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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being   <a href="prob_1.html#NM1" title="PROB_1:NM.1">Field_Subset</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">M</font> being   <a href="measure1.html#NM1" title="MEASURE1:NM.1">Measure</a> of <font color="Olive" title="b1">F</font><br/>  for <font color="Olive" title="b3">Sets</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b4">Cvr</font> being    <a href="measure8.html#M4" title="MEASURE8:mode.4">Covering</a> of <font color="Olive" title="b3">Sets</font>,<font color="Olive" title="b1">F</font> holds   <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> (<font color="Olive" title="b2">M</font>,<span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b3">Sets</font></span>)</span></span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Olive" title="b2">M</font>,<font color="Olive" title="b4">Cvr</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="prob_1.html#NM1" title="PROB_1:NM.1">Field_Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   <a href="measure1.html#NM1" title="MEASURE1:NM.1">Measure</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b2">Sets</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">Cvr</font> being    <a href="measure8.html#M4" title="MEASURE8:mode.4">Covering</a> of <font color="Olive" title="b2">Sets</font>,<font color="Maroon" title="c2">F</font> holds   <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> (<font color="Olive" title="b1">M</font>,<span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">Sets</font></span>)</span></span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Olive" title="b1">M</font>,<font color="Olive" title="b3">Cvr</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM1" title="MEASURE1:NM.1">Measure</a> of <font color="Maroon" title="c2">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Sets</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">Cvr</font> being    <a href="measure8.html#M4" title="MEASURE8:mode.4">Covering</a> of <font color="Olive" title="b1">Sets</font>,<font color="Maroon" title="c2">F</font> holds   <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">Sets</font></span>)</span></span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b2">Cvr</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">Sets</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Cvr</font> being    <a href="measure8.html#M4" title="MEASURE8:mode.4">Covering</a> of <font color="Maroon" title="c4">Sets</font>,<font color="Maroon" title="c2">F</font> holds   <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">Sets</font></span>)</span></span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">Cvr</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">Cvr</font> be    <a href="measure8.html#M4" title="MEASURE8:mode.4">Covering</a> of <font color="Maroon" title="c4">Sets</font>,<font color="Maroon" title="c2">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">Sets</font></span>)</span></span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">Cvr</font>)</span>)</span></span><br/>









<span class="kw">set </span><font color="Maroon" title="c6">Q</font> =  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c5">Cvr</font></span>)</span>)</span>)</span>;<br/>
<a NAME="E1:25"/>
 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#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c5">Cvr</font></span>)</span>)</span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b2">y</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 <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">Cvr</font>)</span>)</span></span>)</span> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c5">Cvr</font></span>)</span>)</span>)</span></span>)</span> implies  ex <font color="Olive" title="b1">y</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 <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">Cvr</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">y</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:25_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c5">Cvr</font></span>)</span>)</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</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 <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">Cvr</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">y</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:25_1"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c5">Cvr</font></span>)</span>)</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:25_1"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
 for <font color="Olive" title="b1">Sets</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">G</font> being    <a href="measure8.html#M4" title="MEASURE8:mode.4">Covering</a> of <font color="Olive" title="b1">Sets</font>,<font color="Maroon" title="c2">F</font> holds  <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Olive" title="b2">G</font></span>)</span>)</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b2">G</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="measure8.html#T6" target="_self" title="MEASURE8:th.6">Th6</a></span>;<br/>
<span class="kw">take </span>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">Cvr</font>)</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c9">m</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">Cvr</font>)</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">Cvr</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">x</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#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">Cvr</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="E6:25_1"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
 for <font color="Olive" title="b1">Sets</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">G</font> being    <a href="measure8.html#M4" title="MEASURE8:mode.4">Covering</a> of <font color="Olive" title="b1">Sets</font>,<font color="Maroon" title="c2">F</font> holds  <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Olive" title="b2">G</font></span>)</span>)</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">n</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#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b2">G</font>)</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c9">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">Sets</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G</font> being    <a href="measure8.html#M4" title="MEASURE8:mode.4">Covering</a> of <font color="Maroon" title="c10">Sets</font>,<font color="Maroon" title="c2">F</font> holds  <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Olive" title="b1">G</font></span>)</span>)</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">n</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#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">G</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">let </span><font color="Maroon" title="c11">G</font> be    <a href="measure8.html#M4" title="MEASURE8:mode.4">Covering</a> of <font color="Maroon" title="c10">Sets</font>,<font color="Maroon" title="c2">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c11">G</font></span>)</span>)</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">n</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#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c11">G</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="E1:25_1_1"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c11">G</font></span>)</span>)</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c11">G</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="txt" href="#E5:25_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a></span>;<br/>
<a NAME="E2:25_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c11">G</font></span>)</span>)</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c11">G</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="measure8.html#T1" target="_self" title="MEASURE8:th.1">Th1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:25_1_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c11">G</font></span>)</span>)</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">n</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#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c11">G</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="measure8.html#T1" target="_self" title="MEASURE8:th.1">Th1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:25_1"/>
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:25_1"/>
( <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">Cvr</font>)</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">Cvr</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">x</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#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">Cvr</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="txt" href="#E3:25_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E6:25_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A4</font></span>: 
 <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c5">Cvr</font></span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">Cvr</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#T63" title="XXREAL_2:th.63">XXREAL_2:63</a></span>;<br/>
<a NAME="E3:25"/>
 <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c5">Cvr</font></span>)</span>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">Sets</font></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D7" target="_self" title="MEASURE8:def.7">Def7</a></span>;<br/>
<a NAME="E4:25"/><span class="kw">then </span>
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">Sets</font></span>)</span></span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c5">Cvr</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#T3" title="XXREAL_2:th.3">XXREAL_2:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:25"/>
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">Sets</font></span>)</span></span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">Cvr</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
