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

<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  <a href="prob_1.html#K14" title="PROB_1:func.14">Borel_Sets</a> ;<br/>

<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E7">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/>

<div><i><font color="Green" title="E8">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">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="E1:11_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E2:11_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">n</font></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/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_2_1"/>
( 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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">ASeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> or  ex <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>  st not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">ASeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2_1_1"><b>suppose </b></a><a NAME="E1:11_2_1_1"/><i><font color="Green" title="E9">A3</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   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">ASeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 ;<br/><div class="add"><div><i><font color="Green" title="E10">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">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="E1:11_2_1_1_1"/><i><font color="Green" title="E11">A5</font></i>: 
<font color="Maroon" title="c3">ASeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K14" title="PROB_1:func.14">Borel_Sets</a> 
 <b>by </b><i><a class="ref" href="prob_1.html#D9" title="PROB_1:def.9">PROB_1:def 9</a></i>;<br/><a NAME="E2:11_2_1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">ASeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="qmax_1.html#E1:11_2_1_1"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/><a NAME="E3:11_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">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="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="qmax_1.html#E1:11_2_1_1_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="txt" href="qmax_1.html#E7"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/><b>hence </b><a NAME="E4:11_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="qmax_1.html#E2:11"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/></div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c4">r</font> = 1 as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E4:11_2_1_1"/>
 ex <font color="Olive" title="b1">m</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>  st <br/>for <font color="Olive" title="b2">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>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">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="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1_1_2"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
;<br/>

<b>thus </b><a NAME="E1:11_2_1_1_2"/>
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>   st  <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="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">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="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="qmax_1.html#E2:11_2_1_1"><i><font color="Green" title="E10">A4</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:11_2_1_1"/><b>then </b><i><font color="Green" title="E11">A6</font></i>: 
( <font color="Maroon" title="c2">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c3">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="c2">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> 1 )
 <b>by </b><i><a class="ref" href="prob_1.html#T3" title="PROB_1:th.3">PROB_1:3</a></i>;<br/><a NAME="E6:11_2_1_1"/><i><font color="Green" title="E12">A7</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c3">ASeq</font>
 <b>by </b><i><a class="txt" href="qmax_1.html#E1:11_2_1_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="prob_1.html#T29" title="PROB_1:th.29">PROB_1:29</a></i>;<br/><a NAME="E7:11_2_1_1"/>
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="c3">ASeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K14" title="PROB_1:func.14">Borel_Sets</a> 
 <b>by </b><i><a class="ref" href="prob_1.html#D9" title="PROB_1:def.9">PROB_1:def 9</a></i>;<br/><a NAME="E8:11_2_1_1"/><b>then </b>
 <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c3">ASeq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K14" title="PROB_1:func.14">Borel_Sets</a> 
 <b>by </b><i><a class="ref" href="prob_1.html#D8" title="PROB_1:def.8">PROB_1:def 8</a></i>;<br/><b>hence </b><a NAME="E9:11_2_1_1"/>
( <font color="Maroon" title="c2">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c3">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="c2">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> <font color="Maroon" title="c2">P</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="c3">ASeq</font></span>)</span> )
 <b>by </b><i><a class="txt" href="qmax_1.html#E5:11_2_1_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="qmax_1.html#E6:11_2_1_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="qmax_1.html#E7"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2_1_2"><b>suppose </b></a><a NAME="E1:11_2_1_2"/><i><font color="Green" title="E9">A8</font></i>: 
not 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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">ASeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 ;<br/><div class="add"><a NAME="E2:11_2_1_2"/>
 ex <font color="Olive" title="b1">m</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>  st <br/>for <font color="Olive" title="b2">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>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">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="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c4">m</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> <b> such that </b><br/><a NAME="E2:11_2_1_2_1"/><i><font color="Green" title="E10">A9</font></i>: 
not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">ASeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="txt" href="qmax_1.html#E1:11_2_1_2"><i><font color="Green" title="E9">A8</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">m</font>
;<br/>

<a NAME="E3:11_2_1_2_1"/>
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>   st <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">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="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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>assume </b><a NAME="E1:11_2_1_2_1_1"/>
<font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font>
 ;<br/>

<a NAME="E2:11_2_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c3">ASeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">ASeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">m</font>
 
<b>by </b><i><a class="txt" href="qmax_1.html#E1:11"><i><font color="Green" title="E7">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="E3:11_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E11">A10</font></i>: 
not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">ASeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
 
<b>by </b><i><a class="txt" href="qmax_1.html#E2:11_2_1_2_1"><i><font color="Green" title="E10">A9</font></i></a></i>;<br/>
<a NAME="E4:11_2_1_2_1_1"/>
<font color="Maroon" title="c3">ASeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K14" title="PROB_1:func.14">Borel_Sets</a> 
 
<b>by </b><i><a class="ref" href="prob_1.html#D9" title="PROB_1:def.9">PROB_1:def 9</a></i>;<br/>
<a NAME="E5:11_2_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c2">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="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="qmax_1.html#E3:11_2_1_2_1_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="txt" href="qmax_1.html#E7"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:11_2_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="qmax_1.html#E2:11"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:11_2_1_2_1"/>
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>   st <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">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="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E3:11_2_1_2"/><b>then </b><i><font color="Green" title="E10">A11</font></i>: 
( <font color="Maroon" title="c2">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c3">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="c2">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#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="prob_1.html#T3" title="PROB_1:th.3">PROB_1:3</a></i>;<br/><a NAME="E4:11_2_1_2"/><i><font color="Green" title="E11">A12</font></i>: 
not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c3">ASeq</font>
 <b>by </b><i><a class="txt" href="qmax_1.html#E1:11_2_1_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="prob_1.html#T29" title="PROB_1:th.29">PROB_1:29</a></i>;<br/><a NAME="E5:11_2_1_2"/>
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="c3">ASeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K14" title="PROB_1:func.14">Borel_Sets</a> 
 <b>by </b><i><a class="ref" href="prob_1.html#D9" title="PROB_1:def.9">PROB_1:def 9</a></i>;<br/><a NAME="E6:11_2_1_2"/><b>then </b>
 <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c3">ASeq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K14" title="PROB_1:func.14">Borel_Sets</a> 
 <b>by </b><i><a class="ref" href="prob_1.html#D8" title="PROB_1:def.8">PROB_1:def 8</a></i>;<br/><b>hence </b><a NAME="E7:11_2_1_2"/>
( <font color="Maroon" title="c2">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c3">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="c2">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> <font color="Maroon" title="c2">P</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="c3">ASeq</font></span>)</span> )
 <b>by </b><i><a class="txt" href="qmax_1.html#E3:11_2_1_2"><i><font color="Green" title="E10">A11</font></i></a>, <a class="txt" href="qmax_1.html#E4:11_2_1_2"><i><font color="Green" title="E11">A12</font></i></a>, <a class="txt" href="qmax_1.html#E7"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:11"/>
( <font color="Maroon" title="c2">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c3">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="c2">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> <font color="Maroon" title="c2">P</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="c3">ASeq</font></span>)</span> )
 ;<br/>


</div>
