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

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="prob_1.html#M1">SetSequence</a> of  <a href="prob_1.html#K14">Borel_Sets</a> ;<br/>

<b>assume </b><a NAME="E1:11"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="prob_1.html#V2">non-increasing</a>
 ;<br/>

<div><i><font color="Green">E8</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">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:11_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</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="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E2:11_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</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>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</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="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T22">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">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds 0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> or ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st not 0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds 0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><div class="add"><div><i><font color="Green">E10</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">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:11_2_1_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K14">Borel_Sets</a> 
 <b>by </b><i><a class="ref" href="prob_1.html#D9">PROB_1:def 9</a></i>;<br/><a NAME="E2:11_2_1_1_1"/>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="qmax_1.html#E1:11_2_1_1"><i><font color="Green">E9</font></i></a></i>;<br/><a NAME="E3:11_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</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="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="qmax_1.html#E1:11_2_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="qmax_1.html#E7"><i><font color="Green">Lemma2</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">c<sub>2</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>4</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="qmax_1.html#E2:11"><i><font color="Green">E8</font></i></a></i>;<br/></div><b>end;</b></div><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = 1 as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E4:11_2_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</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="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></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>
0
;<br/>

<b>thus </b><a NAME="E1:11_2_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="qmax_1.html#E2:11_2_1_1"><i><font color="Green">E10</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:11_2_1_1"/><b>then </b><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</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> 1 )
 <b>by </b><i><a class="ref" href="prob_1.html#T3">PROB_1:3</a></i>;<br/><a NAME="E6:11_2_1_1"/><i><font color="Green">E12</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K4">Intersection</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="qmax_1.html#E1:11_2_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="prob_1.html#T29">PROB_1:29</a></i>;<br/><a NAME="E7:11_2_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K14">Borel_Sets</a> 
 <b>by </b><i><a class="ref" href="prob_1.html#D9">PROB_1:def 9</a></i>;<br/><a NAME="E8:11_2_1_1"/><b>then </b>
 <a href="prob_1.html#K4">Intersection</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K14">Borel_Sets</a> 
 <b>by </b><i><a class="ref" href="prob_1.html#D8">PROB_1:def 8</a></i>;<br/><b>hence </b><a NAME="E9:11_2_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</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> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="qmax_1.html#E5:11_2_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="qmax_1.html#E6:11_2_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="qmax_1.html#E7"><i><font color="Green">Lemma2</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">E9</font></i>: 
not for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds 0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:11_2_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</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="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 <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">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:11_2_1_2_1"/><i><font color="Green">E10</font></i>: 
not 0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="qmax_1.html#E1:11_2_1_2"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<a NAME="E3:11_2_1_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 
<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">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:11_2_1_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:11_2_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="qmax_1.html#E1:11"><i><font color="Green">E7</font></i></a>, <a class="ref" href="prob_1.html#D6">PROB_1:def 6</a></i>;<br/>
<a NAME="E3:11_2_1_2_1_1"/><b>then </b><i><font color="Green">E11</font></i>: 
not 0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="qmax_1.html#E2:11_2_1_2_1"><i><font color="Green">E10</font></i></a></i>;<br/>
<a NAME="E4:11_2_1_2_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K14">Borel_Sets</a> 
 
<b>by </b><i><a class="ref" href="prob_1.html#D9">PROB_1:def 9</a></i>;<br/>
<a NAME="E5:11_2_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</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="prob_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="qmax_1.html#E3:11_2_1_2_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="qmax_1.html#E7"><i><font color="Green">Lemma2</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">c<sub>2</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> 0
 <b>by </b><i><a class="txt" href="qmax_1.html#E2:11"><i><font color="Green">E8</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">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div><b>end;</b></div><a NAME="E3:11_2_1_2"/><b>then </b><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</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> 0 )
 <b>by </b><i><a class="ref" href="prob_1.html#T3">PROB_1:3</a></i>;<br/><a NAME="E4:11_2_1_2"/><i><font color="Green">E11</font></i>: 
not 0 <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K4">Intersection</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="qmax_1.html#E1:11_2_1_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="prob_1.html#T29">PROB_1:29</a></i>;<br/><a NAME="E5:11_2_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K14">Borel_Sets</a> 
 <b>by </b><i><a class="ref" href="prob_1.html#D9">PROB_1:def 9</a></i>;<br/><a NAME="E6:11_2_1_2"/><b>then </b>
 <a href="prob_1.html#K4">Intersection</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K14">Borel_Sets</a> 
 <b>by </b><i><a class="ref" href="prob_1.html#D8">PROB_1:def 8</a></i>;<br/><b>hence </b><a NAME="E7:11_2_1_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</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> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="qmax_1.html#E3:11_2_1_2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="qmax_1.html#E4:11_2_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="qmax_1.html#E7"><i><font color="Green">Lemma2</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">c<sub>2</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</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> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font></span>)</span> )
 ;<br/>


</div>
