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

<b>let </b><font color="Maroon">a</font> be  <a href="xxreal_0.html#V2">positive</a> <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> be  <a href="xxreal_0.html#V2">positive</a> <a href="real_1.html#NM1">Real</a>;<br/>

<b>set </b><font color="Maroon">f</font> =  <a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>] );<br/>
<a NAME="E1:102"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1"><b>proof </b></a><div class="add">

<a NAME="E1:102_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:102_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:102_1_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> 1 holds <br/><span class="p1">(<span class="default"><a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<i><font color="Green">E51</font></i>: <span class="p1">(<span class="default"><a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> 1 = 
 <a href="asympt_1.html#K7" target="_self">Prob28</a> 1,<font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="asympt_1.html#E7:102_1_1_1"><i><font color="Green">E61</font></i></a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i/>
;<br/>
<a NAME="E3:102_1_1_1"/><i><font color="Green">E52</font></i>: 
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="int_1.html#T54">INT_1:54</a></i>;<br/>
<b>consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:102_1_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="int_1.html#K2">\]</a></span>
 <b>and </b><br/><a NAME="E6:102_1_1_1"/><i><font color="Green">E60</font></i>: 
 <a href="asympt_1.html#K7" target="_self">Prob28</a> 2,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K7" target="_self">Prob28</a> <font color="Maroon">n1</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K4">*</a> 2</span>)</span>
 <b>by </b><i/>;<br/>
<i><font color="Green">E61</font></i>: <span class="p1">(<span class="default"><a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K7" target="_self">Prob28</a> 1,<font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K4">*</a> 2</span>)</span>
<b>by </b><i>, , , <a class="txt" href="asympt_1.html#E7:102_1_1_1"><i><font color="Green">E61</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">4 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K4">*</a> 2</span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E8:102_1_1_1"/>
4 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E9:102_1_1_1"/><b>then </b>
<span class="p1">(<span class="default">4 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> <a href="real_1.html#K3">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E10:102_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[1]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:102_1_1"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><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><b>that </b><br/><a NAME="E1:102_1_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>and </b><br/><a NAME="E2:102_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">m</font>]
 ;<br/>

<b>assume </b><a NAME="E3:102_1_1_2"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 holds <br/><span class="p1">(<span class="default"><a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E4:102_1_1_2"/><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>set </b><font color="Maroon">t1</font> = <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="int_1.html#K2">\]</a></span>;<br/>
<b>set </b><font color="Maroon">t2</font> = <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="int_1.html#K2">\]</a></span>;<br/>
<a NAME="E5:102_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E6:102_1_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E7:102_1_1_2"/><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/>
<a NAME="E8:102_1_1_2"/><i><font color="Green">E71</font></i>: 
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="asympt_1.html#E7:102_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t1</font> = <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="int_1.html#K2">\]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="asympt_1.html#E8:102_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E10:102_1_1_2"/>
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">t2</font> = <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="int_1.html#K2">\]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E12:102_1_1_2"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">t2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:102_1_1_2_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 0 or <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T62">NAT_1:62</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:102_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:102_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>;<br/><b>then </b>0 = 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="int_1.html#K5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="int_1.html#K5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> 2</span>)</span></span>)</span>

;<br/><a NAME="E4:102_1_1_2_1_1_1"/><b>then </b>
2 <a href="int_1.html#R2">divides</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/><a NAME="E5:102_1_1_2_1_1_1"/><b>then </b>
2 <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i><a class="ref" href="wsierp_1.html#T9">WSIERP_1:9</a></i>;<br/><b>then reconsider </b><font color="Maroon">t4</font> = <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="real_1.html#K6">/</a> 2 as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="ref" href="wsierp_1.html#T22">WSIERP_1:22</a></i>;<br/><a NAME="E7:102_1_1_2_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">t4</font> <a href="xcmplx_0.html#K6">-</a> 1 is   <a href="int_1.html#NM1">Integer</a>
 ;<br/><a NAME="E8:102_1_1_2_1_1_1"/><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2
 ;<br/><a NAME="E9:102_1_1_2_1_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">t2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/><a NAME="E10:102_1_1_2_1_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">t2</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/><div><i><font color="Green">E77</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_2_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:102_1_1_2_1_1_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">t2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t4</font>
 ;<br/><b>thus </b><a NAME="E2:102_1_1_2_1_1_1_2"/>
contradiction
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_2_1_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1_2_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:102_1_1_2_1_1_1_2_1_1"/>
( <font color="Maroon">t2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">t4</font> or <font color="Maroon">t2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t4</font> )
 </span><b>by </b><i><a class="txt" href="asympt_1.html#E1:102_1_1_2_1_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1_2_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:102_1_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon">t2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">t4</font>
 ;<br/><div class="add"><a NAME="E2:102_1_1_2_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">t2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E3:102_1_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 5 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 4
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E4:102_1_1_2_1_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 5</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E5:102_1_1_2_1_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 5</span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 ;<br/><a NAME="E6:102_1_1_2_1_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 5</span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K6">/</a> 2
 ;<br/><b>hence </b><a NAME="E7:102_1_1_2_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="asympt_1.html#E2:102_1_1_2_1_1_1_2_1_1_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1_2_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:102_1_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon">t2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t4</font>
 ;<br/><div class="add"><a NAME="E2:102_1_1_2_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">t2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t4</font>
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E3:102_1_1_2_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">t2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="asympt_1.html#E9:102_1_1_2_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E4:102_1_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E5:102_1_1_2_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E6:102_1_1_2_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 ;<br/><a NAME="E7:102_1_1_2_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2
 ;<br/><b>hence </b><a NAME="E8:102_1_1_2_1_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="asympt_1.html#E10:102_1_1_2_1_1_1"><i><font color="Green">E75</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><a NAME="E12:102_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">t4</font>
 ;<br/><a NAME="E13:102_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">t1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E8:102_1_1_2_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="asympt_1.html#E11:102_1_1_2_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="int_1.html#T54">INT_1:54</a></i>;<br/><a NAME="E14:102_1_1_2_1_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">t1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E7:102_1_1_2_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E15:102_1_1_2_1_1_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:102_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E16:102_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:102_1_1"><i><font color="Green">E62</font></i></a></i>;<br/><b>hence </b><a NAME="E17:102_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">t2</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:102_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:102_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>;<br/><a NAME="E3:102_1_1_2_1_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/><a NAME="E4:102_1_1_2_1_1_2"/><i><font color="Green">E85</font></i>: 
( 2 <a href="int_1.html#R2">divides</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2 iff <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2 is   <a href="int_1.html#NM1">Integer</a> )
 <b>by </b><i><a class="ref" href="wsierp_1.html#T22">WSIERP_1:22</a></i>;<br/>1 = 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="int_1.html#K5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> 2</span>)</span>
<b>by </b><i>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="int_1.html#K5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> 2</span>)</span></span>)</span>

;<br/><a NAME="E6:102_1_1_2_1_1_2"/><b>then </b>
2 <a href="int_1.html#R2">divides</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/><b>then reconsider </b><font color="Maroon">t3</font> = <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2 as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i>, <a class="ref" href="wsierp_1.html#T9">WSIERP_1:9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_2_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:102_1_1_2_1_1_2_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">t1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t3</font>
 ;<br/><b>thus </b><a NAME="E2:102_1_1_2_1_1_2_2"/>
contradiction
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_2_1_1_2_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1_2_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:102_1_1_2_1_1_2_2_1_1"/>
( <font color="Maroon">t1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">t3</font> or <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t3</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1_2_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:102_1_1_2_1_1_2_2_1_1_1"/>
<font color="Maroon">t1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">t3</font>
 ;<br/><div class="add"><a NAME="E2:102_1_1_2_1_1_2_2_1_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">t1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t3</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E3:102_1_1_2_1_1_2_2_1_1_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 4 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 3
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E4:102_1_1_2_1_1_2_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E5:102_1_1_2_1_1_2_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 ;<br/><a NAME="E6:102_1_1_2_1_1_2_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="real_1.html#K6">/</a> 2
 ;<br/><b>hence </b><a NAME="E7:102_1_1_2_1_1_2_2_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1_2_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:102_1_1_2_1_1_2_2_1_1_2"/>
<font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t3</font>
 ;<br/><div class="add"><a NAME="E2:102_1_1_2_1_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">t1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t3</font>
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E3:102_1_1_2_1_1_2_2_1_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t3</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E4:102_1_1_2_1_1_2_2_1_1_2"/>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E5:102_1_1_2_1_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E6:102_1_1_2_1_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 ;<br/><a NAME="E7:102_1_1_2_1_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2
 ;<br/><b>hence </b><a NAME="E8:102_1_1_2_1_1_2_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="asympt_1.html#E8:102_1_1_2"><i><font color="Green">E71</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:102_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">t2</font>
 <b>by </b><i><a class="ref" href="int_1.html#T54">INT_1:54</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E13:102_1_1_2"/><b>then </b><i><font color="Green">E89</font></i>: 
4 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">t1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 4 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">t2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E14:102_1_1_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">b</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_1.html#E6:102_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E15:102_1_1_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="asympt_1.html#E4:102_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E17:102_1_1_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="int_1.html#K2">\]</a></span>
 <b>and </b><br/><a NAME="E18:102_1_1_2"/><i><font color="Green">E93</font></i>: 
 <a href="asympt_1.html#K7" target="_self">Prob28</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K7" target="_self">Prob28</a> <font color="Maroon">n1</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:102_1_1_2"><i><font color="Green">E63</font></i></a></i>;<br/>
<i><font color="Green">E94</font></i>: <span class="p1">(<span class="default"><a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K7" target="_self">Prob28</a> <font color="Maroon">n1</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>
<a NAME="E20:102_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i><a class="txt" href="asympt_1.html#E6:102_1_1_2"><i><font color="Green">E69</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>then consider </b><font color="Maroon">n2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E22:102_1_1_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="int_1.html#K2">\]</a></span>
 <b>and </b><br/><a NAME="E23:102_1_1_2"/><i><font color="Green">E96</font></i>: 
 <a href="asympt_1.html#K7" target="_self">Prob28</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K7" target="_self">Prob28</a> <font color="Maroon">n2</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:102_1_1_2"><i><font color="Green">E63</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K7" target="_self">Prob28</a> <font color="Maroon">n2</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">t2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E25:102_1_1_2"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:102_1_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S9">NAT_1:sch 9</a>(, );</i><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:102_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:102"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S10">NAT_1:sch 10</a>();<br/></i>
<b>hence </b><a NAME="E3:102"/>
 <a href="asympt_1.html#K8" target="_self">seq_prob28</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> is <a href="asympt_0.html#V6">eventually-nondecreasing</a>
 <b>by </b><i><a class="ref" href="asympt_0.html#D8">ASYMPT_0:def 8</a></i>;<br/>


</div>
