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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">M</font> be    <a href="measure1.html#M3" title="MEASURE1:mode.3">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>;<br/>





<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 1
 ;<br/>

<a NAME="E2:21"/><i><font color="Green" title="E18">A2</font></i>: 
for <font color="Olive" title="b1">A</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> holds  <font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">A</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">X</font> = <font color="Maroon" title="c1">X</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="prob_1.html#T11" title="PROB_1:th.11">PROB_1:11</a></i>;<br/>
<a NAME="E2:21_1"/>
<font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c4">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c5">X</font>
 
<b>by </b><i><a class="ref" href="measure1.html#T62" title="MEASURE1:th.62">MEASURE1:62</a></i>;<br/>
<b>hence </b><a NAME="E3:21_1"/>
<font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c4">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 1
 <b>by </b><i><a class="txt" href="prob_4.html#E1:21"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:21"/><i><font color="Green" title="E19">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:21"/><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon" title="c3">M</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="measure1.html#D11" title="MEASURE1:def.11">MEASURE1:def 11</a></i>;<br/>
<a NAME="E5:21"/><i><font color="Green" title="E21">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> holds <br/><font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:21_2"/><i><font color="Green" title="E22">A6</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="prob_4.html#E1:21_2"><i><font color="Green" title="E22">A6</font></i></a></i>;<br/>
<a NAME="E3:21_2"/><i><font color="Green" title="E23">A7</font></i>: 
(  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 1 is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> )
 
<b>by </b><i><a class="ref" href="measure6.html#D1" title="MEASURE6:def.1">MEASURE6:def 1</a></i>;<br/>
<a NAME="E4:21_2"/><i><font color="Green" title="E24">A8</font></i>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c5">x</font>
 
<b>by </b><i><a class="txt" href="prob_4.html#E4:21"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="measure1.html#D4" title="MEASURE1:def.4">MEASURE1:def 4</a></i>;<br/>
<a NAME="E5:21_2"/>
<font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 1
 
<b>by </b><i><a class="txt" href="prob_4.html#E2:21"><i><font color="Green" title="E18">A2</font></i></a></i>;<br/>
<a NAME="E6:21_2"/><b>then </b>
<font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c5">x</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<b>by </b><i><a class="txt" href="prob_4.html#E3:21_2"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="prob_4.html#E4:21_2"><i><font color="Green" title="E24">A8</font></i></a>, <a class="ref" href="measure6.html#T16" title="MEASURE6:th.16">MEASURE6:16</a></i>;<br/>
<b>hence </b><a NAME="E7:21_2"/>
<font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">P1</font> = <font color="Maroon" title="c3">M</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">S</font>, <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  <b>by </b><i><a class="txt" href="prob_4.html#E3:21"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">P1</font> = <font color="Maroon" title="c4">P1</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">S</font>, <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  ;<br/>
<a NAME="E8:21"/><i><font color="Green" title="E22">A9</font></i>: 
for <font color="Olive" title="b1">A</font> being   <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">S</font> holds   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Olive" title="b1">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">A</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">S</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c6">A</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="prob_1.html#D10" title="PROB_1:def.10">PROB_1:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">A1</font> = <font color="Maroon" title="c7">A</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<a NAME="E3:21_3"/><i><font color="Green" title="E23">A10</font></i>: 
( <font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c8">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 
<b>by </b><i><a class="txt" href="prob_4.html#E5:21"><i><font color="Green" title="E21">A5</font></i></a></i>;<br/>
<a NAME="E4:21_3"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c8">A1</font>
 
<b>by </b><i><a class="txt" href="prob_4.html#E4:21"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="measure1.html#D4" title="MEASURE1:def.4">MEASURE1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">p</font>, <font color="Maroon" title="c10">q</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E6:21_3"/><i><font color="Green" title="E24">A11</font></i>: 
( <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c8">A1</font> &amp; <font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">q</font> )
 <b>by </b><i><a class="txt" href="prob_4.html#E3:21_3"><i><font color="Green" title="E23">A10</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:21_3"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c6">A</font>
 <b>by </b><i><a class="txt" href="prob_4.html#E6:21_3"><i><font color="Green" title="E24">A11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:21"/><i><font color="Green" title="E23">A12</font></i>: 
<font color="Maroon" title="c5">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="prob_4.html#E1:21"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="measure6.html#D1" title="MEASURE6:def.1">MEASURE6:def 1</a></i>;<br/>
<a NAME="E10:21"/><i><font color="Green" title="E24">A13</font></i>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\/</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Olive" title="b2">B</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">A</font>, <font color="Maroon" title="c7">B</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">S</font>;<br/>

<b>assume </b><a NAME="E1:21_4"/><i><font color="Green" title="E25">A14</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">B</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">A</font> = <font color="Maroon" title="c6">A</font>, <font color="Maroon" title="c9">B</font> = <font color="Maroon" title="c7">B</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="prob_1.html#D10" title="PROB_1:def.10">PROB_1:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">A2</font> = <font color="Maroon" title="c8">A</font>, <font color="Maroon" title="c11">B2</font> = <font color="Maroon" title="c9">B</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<font color="Maroon" title="c5">P1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c9">B</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c10">A2</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c11">B2</font></span>)</span>
<b>by </b><i><a class="txt" href="prob_4.html#E1:21_4"><i><font color="Green" title="E25">A14</font></i></a>, <a class="ref" href="measure1.html#T61" title="MEASURE1:th.61">MEASURE1:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">P1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">B</font></span>)</span>
<b>by </b><i><a class="ref" href="supinf_2.html#T1" title="SUPINF_2:th.1">SUPINF_2:1</a></i>
;<br/>
<b>hence </b><a NAME="E5:21_4"/>
<font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\/</a> <font color="Maroon" title="c7">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c7">B</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:21"/>
for <font color="Olive" title="b1">ASeq</font> being   <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">ASeq</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-increasing</a> holds <br/>( <font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Olive" title="b1">ASeq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Olive" title="b1">ASeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Olive" title="b1">ASeq</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">ASeq</font> be    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of <font color="Maroon" title="c2">S</font>;<br/>

<b>assume </b><a NAME="E1:21_5"/><i><font color="Green" title="E25">A15</font></i>: 
<font color="Maroon" title="c6">ASeq</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-increasing</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c6">ASeq</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> ,<font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="prob_4.html#T2" target="_self" title="PROB_4:th.2">Th2</a></i>;<br/>
<a NAME="E3:21_5"/><i><font color="Green" title="E26">A16</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</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> <font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>
 
<b>by </b><i><a class="txt" href="prob_4.html#E1:21_5"><i><font color="Green" title="E25">A15</font></i></a>, <a class="ref" href="prob_2.html#T14" title="PROB_2:th.14">PROB_2:14</a></i>;<br/>
<a NAME="E4:21_5"/><i><font color="Green" title="E27">A17</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:21_5_1"/>
<font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 
<b>by </b><i><a class="txt" href="prob_4.html#E5:21"><i><font color="Green" title="E21">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:21_5_1"/>
<font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:21_5"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c6">ASeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">A</font> = <font color="Maroon" title="c6">ASeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c8">n</font> as    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="prob_1.html#D10" title="PROB_1:def.10">PROB_1:def 10</a></i>;<br/>
<a NAME="E2:21_5_2"/><i><font color="Green" title="E28">A18</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c9">A</font>
 
<b>by </b><i><a class="txt" href="prob_4.html#E8:21"><i><font color="Green" title="E22">A9</font></i></a></i>;<br/>
<a NAME="E3:21_5_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c6">ASeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="seq_1.html#T3" title="SEQ_1:th.3">SEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:21_5_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c6">ASeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font>
 <b>by </b><i><a class="txt" href="prob_4.html#E2:21_5_2"><i><font color="Green" title="E28">A18</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:21_5"/><b>then </b><i><font color="Green" title="E28">A19</font></i>: 
<font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c6">ASeq</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T10" title="RINFSUP1:th.10">RINFSUP1:10</a></i>;<br/>
<a NAME="E7:21_5"/><i><font color="Green" title="E29">A20</font></i>: 
<font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c6">ASeq</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:21_5_3_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E2:21_5_3_1"/><b>then </b><i><font color="Green" title="E30">A21</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><a NAME="E3:21_5_3_1"/><b>then </b><i><font color="Green" title="E31">A22</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c8">n</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> )
 <b>by </b><i><a class="txt" href="prob_4.html#E5:21"><i><font color="Green" title="E21">A5</font></i></a></i>;<br/><a NAME="E4:21_5_3_1"/>
<font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</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> <font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font>
 <b>by </b><i><a class="txt" href="prob_4.html#E1:21_5"><i><font color="Green" title="E25">A15</font></i></a>, <a class="ref" href="prob_2.html#T14" title="PROB_2:th.14">PROB_2:14</a></i>;<br/><a NAME="E5:21_5_3_1"/><b>then </b>
<font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="measure1.html#T62" title="MEASURE1:th.62">MEASURE1:62</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">p</font>, <font color="Maroon" title="c10">q</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E7:21_5_3_1"/><i><font color="Green" title="E32">A23</font></i>: 
( <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c8">n</font> &amp; <font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">q</font> )
 <b>by </b><i><a class="txt" href="prob_4.html#E2:21_5_3_1"><i><font color="Green" title="E30">A21</font></i></a>, <a class="txt" href="prob_4.html#E3:21_5_3_1"><i><font color="Green" title="E31">A22</font></i></a></i>;<br/><b>thus </b><a NAME="E8:21_5_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c6">ASeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="c5">P1</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c6">ASeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font>
 <b>by </b><i><a class="txt" href="prob_4.html#E7:21_5_3_1"><i><font color="Green" title="E32">A23</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:21_5_3"/>
<font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c6">ASeq</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D14" title="SEQM_3:def.14">SEQM_3:def 14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:21_5"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c6">ASeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5_4"><b>proof </b></a><div class="add">

<b>thus </b> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c6">ASeq</font></span>)</span> = 
 <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c6">ASeq</font></span>)</span>
<b>by </b><i><a class="txt" href="prob_4.html#E6:21_5"><i><font color="Green" title="E28">A19</font></i></a>, <a class="txt" href="prob_4.html#E7:21_5"><i><font color="Green" title="E29">A20</font></i></a>, <a class="ref" href="rinfsup1.html#T25" title="RINFSUP1:th.25">RINFSUP1:25</a></i>
<br/>.= 
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prob_4.html#E6:21_5"><i><font color="Green" title="E28">A19</font></i></a>, <a class="ref" href="prob_4.html#T12" target="_self" title="PROB_4:th.12">Th12</a></i>
;<br/>


</div><b>end;</b></div>
<b>then </b> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c6">ASeq</font></span>)</span> = 
<font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p2">(<span class="default"><a href="measure1.html#K5" title="MEASURE1:func.5">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prob_4.html#E3:21_5"><i><font color="Green" title="E26">A16</font></i></a>, <a class="txt" href="prob_4.html#E4:21_5"><i><font color="Green" title="E27">A17</font></i></a>, <a class="ref" href="measure3.html#T14" title="MEASURE3:th.14">MEASURE3:14</a></i>
<br/>.= 
<font color="Maroon" title="c5">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c6">ASeq</font></span>)</span>
<b>by </b><i><a class="ref" href="setlim_1.html#T13" title="SETLIM_1:th.13">SETLIM_1:13</a></i>
;<br/>
<b>hence </b><a NAME="E10:21_5"/>
( <font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c6">ASeq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P1</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c6">ASeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c6">ASeq</font></span>)</span> )
 <b>by </b><i><a class="txt" href="prob_4.html#E6:21_5"><i><font color="Green" title="E28">A19</font></i></a>, <a class="txt" href="prob_4.html#E7:21_5"><i><font color="Green" title="E29">A20</font></i></a>, <a class="ref" href="seq_4.html#T52" title="SEQ_4:th.52">SEQ_4:52</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:21"/>
<font color="Maroon" title="c3">M</font> is    <a href="prob_1.html#M3" title="PROB_1:mode.3">Probability</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="prob_4.html#E8:21"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="prob_4.html#E9:21"><i><font color="Green" title="E23">A12</font></i></a>, <a class="txt" href="prob_4.html#E10:21"><i><font color="Green" title="E24">A13</font></i></a>, <a class="ref" href="prob_1.html#D13" title="PROB_1:def.13">PROB_1:def 13</a></i>;<br/>


</div>
