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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="prob_1.html#M3">Probability</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E1:8"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="prob_1.html#V2">non-increasing</a>
 ;<br/>

<a NAME="E2:8"/><i><font color="Green">E10</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>3</sub></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="8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:8_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:8_1"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font> is    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>6</sub></font> is    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="prob_1.html#T57">PROB_1:57</a></i>;<br/><a NAME="E3:8_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="prob_3.html#E1:8_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="prob_3.html#E1:8"><i><font color="Green">E9</font></i></a>, <a class="ref" href="prob_1.html#D6">PROB_1:def 6</a></i>;<br/><a NAME="E4:8_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="prob_3.html#E2:8"><i><font color="Green">E10</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><b>hence </b><a NAME="E5:8_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="prob_3.html#E3:8_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="prob_3.html#E2:8_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="prob_1.html#T70">PROB_1:70</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:8"/>
<font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font> is <a href="seqm_3.html#V4">non-increasing</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#T14">SEQM_3:14</a></i>;<br/>


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