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

<b>consider </b><font color="Maroon">n0</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="E2:66_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n0</font>
 <b>and </b><br/><a NAME="E3:66_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">n0</font>]
 <b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; ( for <font color="Olive">m</font>, <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">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>3</sub></font> holds <br/>( <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">k</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">k</font>] ) ) );<br/>
<a NAME="E4:66_1"/><i><font color="Green">E54</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> <br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_1"><b>suppose </b></a><a NAME="E1:66_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">mx</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>consider </b><font color="Maroon">my</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="E4:66_1_1_1_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">mx</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">my</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">my</font>] )
 <b>by </b><i/>;<br/><b>take </b>
<font color="Maroon">my</font>
;<br/><b>thus </b><a NAME="E5:66_1_1_1_1"/>
<font color="Maroon">my</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>thus </b><a NAME="E6:66_1_1_1_1"/>
for <font color="Olive">m</font>, <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">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">my</font> holds <br/>( <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">k</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">k</font>] )
 <b>by </b><i><a class="txt" href="kurato_2.html#E1:66"><i><font color="Green">E43</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_2"><b>suppose </b></a><a NAME="E1:66_1_1_1_2"/><i><font color="Green">E57</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon">y</font> = 0;<br/><b>take </b>
0
;<br/><b>thus </b><a NAME="E2:66_1_1_1_2"/>
0 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><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> , <font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E3:66_1_1_1_2"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><b>hence </b><a NAME="E4:66_1_1_1_2"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">k</font>] )
 <b>by </b><i><a class="txt" href="kurato_2.html#E1:66"><i><font color="Green">E43</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:66_1"/><i><font color="Green">E72</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E7:66_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">n0</font>
 <b>and </b><br/><a NAME="E8:66_1"/><i><font color="Green">E82</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="lattice5.html#S1">LATTICE5:sch 1</a>();<br/></i>
<a NAME="E9:66_1"/><i><font color="Green">E83</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:66_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:66_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:66_1_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">x</font> 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="kurato_2.html#E1:66"><i><font color="Green">E43</font></i></a>, </i>;<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><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> ;<br/>
<a NAME="E6:66_1_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="ref" href="kurato_2.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E7:66_1_2"/><i><font color="Green">E87</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="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<b>by </b><i/>;<br/>
<a NAME="E8:66_1_2"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="kurato_2.html#E3:66_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="kurato_2.html#T4" target="_self">Th4</a>);<br/></i>
<a NAME="E9:66_1_2"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<b>hence </b><a NAME="E10:66_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E10:66_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="kurato_2.html#E1:66"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="seq_1.html#NM1">Real_Sequence</a> ;<br/>
<a NAME="E13:66_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:66_1_3"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E1:66"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E3:66_1_3"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E1:66"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E5:66_1_3"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:66_1_3"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a> <b>by </b><i>, <a class="ref" href="seqm_3.html#D8">SEQM_3:def 8</a>, <a class="ref" href="seqm_3.html#D11">SEQM_3:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">S1</font> = <font color="Maroon">F<sub>2</sub></font>() <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font> as   <a href="kurato_2.html#NM1">SetSequence</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i/>;<br/>
<a NAME="E16:66_1"/><i><font color="Green">E122</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">S1</font> <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>reconsider </b><font color="Maroon">S1</font> = <font color="Maroon">S1</font> as    <a href="kurato_2.html#M1" target="_self">subsequence</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">S1</font>
;<br/>

<b>thus </b><a NAME="E18:66_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">S1</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">n</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_4_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_4_1_1"><b>suppose </b></a><a NAME="E1:66_1_4_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_1_4_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">S1</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">n</font>]
 <b>by </b><i>, <a class="ref" href="kurato_2.html#T3" target="_self">Th3</a>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_4_1_2"><b>suppose </b></a><a NAME="E1:66_1_4_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:66_1_4_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E3:66_1_4_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then reconsider </b><font color="Maroon">n'</font> = <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> 1 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#T18">INT_1:18</a></i>;<br/><a NAME="E5:66_1_4_1_2"/><i><font color="Green">E124</font></i>: 
for <font color="Olive">m</font>, <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">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n'</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">k</font>]
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">g</font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E7:66_1_4_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>() <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">k</font>]
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:66_1_4_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">S1</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">n</font>]
 <b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>


</div>
