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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">M</font> be    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">S</font>;<br/>





<b>assume </b><a NAME="E1:21"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> 1
 ;<br/>

<a NAME="E2:21"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">A</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> holds  <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <font color="Olive">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">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">A</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font>;<br/>

<b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="prob_1.html#T11">PROB_1:11</a></i>;<br/>
<a NAME="E2:21_1"/>
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/>
<b>hence </b><a NAME="E3:21_1"/>
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> 1
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:21"/><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:21"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">M</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="measure1.html#D11">MEASURE1:def 11</a></i>;<br/>
<a NAME="E5:21"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:21_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E3:21_2"/><i><font color="Green">E51</font></i>: 
(  <a href="measure6.html#K1">R_EAL</a> 1 is   <a href="real_1.html#NM1">Real</a> &amp;  <a href="supinf_2.html#K1">0.</a>  is   <a href="real_1.html#NM1">Real</a> )
 
<b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/>
<a NAME="E4:21_2"/><i><font color="Green">E52</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="measure1.html#D4">MEASURE1:def 4</a></i>;<br/>
<a NAME="E5:21_2"/>
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E6:21_2"/><b>then </b>
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <font color="Maroon">x</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i>, <a class="txt" href="prob_4.html#E1:21"><i><font color="Green">E24</font></i></a>, <a class="ref" href="measure6.html#T16">MEASURE6:16</a></i>;<br/>
<b>hence </b><a NAME="E7:21_2"/>
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">P1</font> = <font color="Maroon">M</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">S</font>, <a href="supinf_1.html#K3">REAL</a>  <b>by </b><i>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P1</font> = <font color="Maroon">P1</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">S</font>, <a href="supinf_1.html#K3">REAL</a>  ;<br/>
<a NAME="E8:21"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">A</font> being   <a href="prob_1.html#M2">Event</a> of <font color="Maroon">S</font> holds  0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">P1</font> <a href="prob_1.html#K10">.</a> <font color="Olive">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">A</font> be    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">S</font>;<br/>

<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="prob_1.html#D10">PROB_1:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">A</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E3:21_3"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <font color="Maroon">A1</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp;  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 
<b>by </b><i>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/>
<a NAME="E4:21_3"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <font color="Maroon">A1</font>
 
<b>by </b><i>, <a class="ref" href="measure1.html#D4">MEASURE1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">q</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:21_3"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E7:21_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">P1</font> <a href="prob_1.html#K10">.</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a>, <a class="txt" href="prob_4.html#E1:21"><i><font color="Green">E24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:21"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/>
<a NAME="E10:21"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font> being   <a href="prob_1.html#M2">Event</a> of <font color="Maroon">S</font>  st <font color="Olive">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> holds <br/><font color="Maroon">P1</font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><font color="Olive">A</font> <a href="prob_1.html#K7">\/</a> <font color="Olive">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="prob_1.html#K10">.</a> <font color="Olive">A</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="prob_1.html#K10">.</a> <font color="Olive">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">A</font> be    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">S</font>, <font color="Maroon">B</font> be    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E1:21_4"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font>, <font color="Maroon">B</font> = <font color="Maroon">B</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="prob_1.html#D10">PROB_1:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A2</font> = <font color="Maroon">A</font>, <font color="Maroon">B2</font> = <font color="Maroon">B</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> ;<br/>
<font color="Maroon">P1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <font color="Maroon">A2</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <font color="Maroon">B2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="measure1.html#T61">MEASURE1:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>
;<br/>
<b>hence </b><a NAME="E5:21_4"/>
<font color="Maroon">P1</font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="prob_1.html#K7">\/</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="prob_1.html#K10">.</a> <font color="Maroon">A</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="prob_1.html#K10">.</a> <font color="Maroon">B</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:21"/>
for <font color="Olive">ASeq</font> being   <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">S</font>  st <font color="Olive">ASeq</font> is <a href="prob_1.html#V2">non-increasing</a> holds <br/>( <font color="Maroon">P1</font> <a href="prob_1.html#K9">*</a> <font color="Olive">ASeq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="prob_1.html#K9">*</a> <font color="Olive">ASeq</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K4">Intersection</a> <font color="Olive">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">ASeq</font> be    <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E1:21_5"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">ASeq</font> is <a href="prob_1.html#V2">non-increasing</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">ASeq</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E3:21_5"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>
 
<b>by </b><i>, <a class="ref" href="prob_2.html#T14">PROB_2:14</a></i>;<br/>
<a NAME="E4:21_5"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+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">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:21_5_1"/>
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:21_5"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">ASeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Maroon">n</font> as    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="prob_1.html#D10">PROB_1:def 10</a></i>;<br/>
<a NAME="E2:21_5_2"/><i><font color="Green">E97</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">P1</font> <a href="prob_1.html#K10">.</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:21_5_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="seq_1.html#T3">SEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:21_5_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T22">FUNCT_1:22</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:21_5"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">P1</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font> is <a href="seq_2.html#V2">bounded_below</a>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T10">RINFSUP1:10</a></i>;<br/>
<a NAME="E7:21_5"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">P1</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font> is <a href="seqm_3.html#V4">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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:21_5_3_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E2:21_5_3_1"/><b>then </b><i><font color="Green">E100</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E3:21_5_3_1"/><b>then </b><i><font color="Green">E101</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is   <a href="real_1.html#NM1">Real</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> is   <a href="real_1.html#NM1">Real</a> )
 <b>by </b><i/>;<br/><a NAME="E4:21_5_3_1"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="prob_2.html#T14">PROB_2:14</a></i>;<br/><a NAME="E5:21_5_3_1"/><b>then </b>
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/><b>then consider </b><font color="Maroon">p</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">q</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:21_5_3_1"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> )
 <b>by </b><i>, <a class="txt" href="prob_4.html#E1"><i><font color="Green">Lemma31</font></i></a></i>;<br/><b>thus </b><a NAME="E8:21_5_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:21_5_3"/>
<font color="Maroon">P1</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font> is <a href="seqm_3.html#V4">non-increasing</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D14">SEQM_3:def 14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:21_5"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">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">lim</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> = 
 <a href="rinfsup1.html#K2">inf</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="rinfsup1.html#T25">RINFSUP1:25</a></i>
<br/>.= 
 <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div>
<b>then </b> <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> = 
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <span class="p2">(<span class="default"><a href="measure1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="measure3.html#T14">MEASURE3:14</a></i>
<br/>.= 
<font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K4">Intersection</a> <font color="Maroon">ASeq</font></span>)</span>
<b>by </b><i><a class="ref" href="setlim_1.html#T13">SETLIM_1:13</a></i>
;<br/>
<b>hence </b><a NAME="E10:21_5"/>
( <font color="Maroon">P1</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K4">Intersection</a> <font color="Maroon">ASeq</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="seq_4.html#T52">SEQ_4:52</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:21"/>
<font color="Maroon">M</font> is    <a href="prob_1.html#M3">Probability</a> of <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="txt" href="prob_4.html#E1:21"><i><font color="Green">E24</font></i></a>, <a class="ref" href="prob_1.html#D13">PROB_1:def 13</a></i>;<br/>


</div>
