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

<b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>thus </b><a NAME="E1:39"/>
( <font color="Maroon">seq</font> is  <a href="seqm_3.html#V1" target="_self">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a> implies ( <font color="Maroon">seq</font> is <a href="seqm_3.html#V1" target="_self">increasing</a> &amp; ( 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">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">seq</font> is  <a href="seqm_3.html#V1" target="_self">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a>
 ;<br/>

<b>hence </b><a NAME="E2:39_1"/>
<font color="Maroon">seq</font> is <a href="seqm_3.html#V1" target="_self">increasing</a>
 ;<br/>

<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="E3:39_1"/><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:39_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<a NAME="E5:39_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">seq</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:39_1"/><b>then </b>
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">seq</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:39_1"/>
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="seqm_3.html#E3:39"><i><font color="Green">E37</font></i></a></i>;<br/>


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

<b>assume </b><b>that </b><br/><a NAME="E2:39"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">seq</font> is <a href="seqm_3.html#V1" target="_self">increasing</a>
 <b>and </b><br/><a NAME="E3:39"/><i><font color="Green">E37</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">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<a NAME="E4:39"/>
<font color="Maroon">seq</font> is <a href="seqm_3.html#V6" target="_self">natural-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:39_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">seq</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="E3:39_2_1"/><i><font color="Green">E38</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">seq</font>
 <b>and </b><br/><a NAME="E4:39_2_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</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/><a NAME="E5:39_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="seqm_3.html#E3:39_2_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E6:39_2_1"/><b>then </b>
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="seqm_3.html#E3:39"><i><font color="Green">E37</font></i></a></i>;<br/><b>hence </b><a NAME="E7:39_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="seqm_3.html#E4:39_2_1"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:39_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="seqm_3.html#D8" target="_self">SEQM_3:def 8</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:39"/>
<font color="Maroon">seq</font> is  <a href="seqm_3.html#V1" target="_self">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a>
 <b>by </b><i/>;<br/>


</div>
