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

<b>let </b><font color="Maroon" title="c1">Omega</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">Sigma</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">Omega</font>;<br/><b>let </b><font color="Maroon" title="c3">ASeq</font> be    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font>;<br/><b>let </b><font color="Maroon" title="c4">P</font> be    <a href="prob_1.html#M3" title="PROB_1:mode.3">Probability</a> of <font color="Maroon" title="c2">Sigma</font>;<br/>







<b>assume </b><a NAME="E1:8"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c3">ASeq</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-increasing</a>
 ;<br/>

<a NAME="E2:8"/><i><font color="Green" title="E10">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c3">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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">n</font>, <font color="Maroon" title="c6">m</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>assume </b><a NAME="E1:8_1"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font>
 ;<br/><a NAME="E2:8_1"/><i><font color="Green" title="E12">A4</font></i>: 
( <font color="Maroon" title="c3">ASeq</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c5">n</font> is    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">Sigma</font> &amp; <font color="Maroon" title="c3">ASeq</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c6">m</font> is    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">Sigma</font> )
 <b>by </b><i><a class="ref" href="prob_1.html#T57" title="PROB_1:th.57">PROB_1:57</a></i>;<br/><a NAME="E3:8_1"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c3">ASeq</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c6">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">ASeq</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="prob_3.html#E1:8_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="prob_3.html#E1:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="prob_1.html#D6" title="PROB_1:def.6">PROB_1:def 6</a></i>;<br/><a NAME="E4:8_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c3">ASeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ASeq</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c5">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c3">ASeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ASeq</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c6">m</font></span>)</span> )
 <b>by </b><i><a class="txt" href="prob_3.html#E2:8"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><b>hence </b><a NAME="E5:8_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c3">ASeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c3">ASeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="prob_3.html#E3:8_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="prob_3.html#E2:8_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="prob_1.html#T70" title="PROB_1:th.70">PROB_1:70</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:8"/>
<font color="Maroon" title="c4">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c3">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#T14" title="SEQM_3:th.14">SEQM_3:14</a></i>;<br/>


</div>
