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<b>let </b><font color="Maroon">Omega</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">Sigma</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font>;<br/><b>let </b><font color="Maroon">ASeq</font> be    <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">Sigma</font>;<br/><b>let </b><font color="Maroon">P</font> be    <a href="prob_1.html#M3">Probability</a> of <font color="Maroon">Sigma</font>;<br/>







<b>assume </b><a NAME="E1:7"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">ASeq</font> is <a href="prob_1.html#V3">non-decreasing</a>
 ;<br/>

<a NAME="E2:7"/><i><font color="Green">E52</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_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> , <font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:7_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:7_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">ASeq</font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">n</font> is    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">Sigma</font> &amp; <font color="Maroon">ASeq</font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">m</font> is    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">Sigma</font> )
 <b>by </b><i><a class="ref" href="prob_1.html#T57">PROB_1:57</a></i>;<br/><a NAME="E3:7_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">ASeq</font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">ASeq</font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="ref" href="prob_1.html#D7">PROB_1:def 7</a></i>;<br/><a NAME="E4:7_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">P</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> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ASeq</font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">P</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">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ASeq</font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">m</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><b>hence </b><a NAME="E5:7_1"/>
<span class="p1">(<span class="default"><font color="Maroon">P</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> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">P</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">m</font>
 <b>by </b><i>, , <a class="ref" href="prob_1.html#T70">PROB_1:70</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:7"/>
<font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font> is <a href="seqm_3.html#V3">non-decreasing</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#T12">SEQM_3:12</a></i>;<br/>


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