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

<b>let </b><font color="Maroon">x0</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>



<b>thus </b><a NAME="E1:18"/>
( <font color="Maroon">f</font> <a href="limfunc3.html#R3" target="_self">is_divergent_to-infty_in</a> <font color="Maroon">x0</font> implies ( <font color="Maroon">f</font> <a href="limfunc2.html#R3">is_left_divergent_to-infty_in</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">f</font> <a href="limfunc2.html#R6">is_right_divergent_to-infty_in</a> <font color="Maroon">x0</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">f</font> <a href="limfunc3.html#R3" target="_self">is_divergent_to-infty_in</a> <font color="Maroon">x0</font>
 ;<br/>

<a NAME="E2:18_1"/><b>then </b><i><font color="Green">E38</font></i>: 
for <font color="Olive">r1</font>, <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r2</font> holds <br/> ex <font color="Olive">g1</font>, <font color="Olive">g2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g1</font> &amp; <font color="Olive">g1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Olive">g1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">g2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r2</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g2</font> &amp; <font color="Olive">g2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:18_1"/><b>then </b><i><font color="Green">E39</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> holds <br/> ex <font color="Olive">g</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g</font> &amp; <font color="Olive">g</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:18_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">s</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">x0</font></span>)</span> )
 ;<br/><a NAME="E2:18_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:18_1_1"/>
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">s</font> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:18_1"/>
<font color="Maroon">f</font> <a href="limfunc2.html#R3">is_left_divergent_to-infty_in</a> <font color="Maroon">x0</font>
 <b>by </b><i>, <a class="ref" href="limfunc2.html#D3">LIMFUNC2:def 3</a></i>;<br/>

<a NAME="E6:18_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> holds <br/> ex <font color="Olive">g</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">g</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:18_1_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">s</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span> )
 ;<br/><a NAME="E2:18_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:18_1_2"/>
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">s</font> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:18_1"/>
<font color="Maroon">f</font> <a href="limfunc2.html#R6">is_right_divergent_to-infty_in</a> <font color="Maroon">x0</font>
 <b>by </b><i>, <a class="ref" href="limfunc2.html#D6">LIMFUNC2:def 6</a></i>;<br/>


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

<b>assume </b><a NAME="E2:18"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">f</font> <a href="limfunc2.html#R3">is_left_divergent_to-infty_in</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">f</font> <a href="limfunc2.html#R6">is_right_divergent_to-infty_in</a> <font color="Maroon">x0</font> )
 ;<br/>

<div><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">r2</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:18_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> )
 ;<br/><b>then consider </b><font color="Maroon">g1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:18_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="ref" href="limfunc2.html#D3">LIMFUNC2:def 3</a></i>;<br/><b>consider </b><font color="Maroon">g2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:18_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">g2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i>, , <a class="ref" href="limfunc2.html#D6">LIMFUNC2:def 6</a></i>;<br/><b>take </b><font color="Maroon">g1</font> = <font color="Maroon">g1</font>;<br/><b>take </b><font color="Maroon">g2</font> = <font color="Maroon">g2</font>;<br/><b>thus </b><a NAME="E6:18_2"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:18_3"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">s</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_3_1_1"/>
(  ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>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">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> or for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x0</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_3_1_1_1"><b>suppose </b></a><a NAME="E1:18_3_1_1_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>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">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</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="E3:18_3_1_1_1"/><i><font color="Green">E67</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="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font>
 ;<br/><a NAME="E4:18_3_1_1_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</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">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> )
 <b>by </b><i>, <a class="ref" href="seq_4.html#T33">SEQ_4:33</a></i>;<br/><a NAME="E5:18_3_1_1_1"/><i><font color="Green">E69</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:18_3_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">x0</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:18_3_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">n</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="E3:18_3_1_1_1_1"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/>
<a NAME="E4:18_3_1_1_1_1"/>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/>
<a NAME="E5:18_3_1_1_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/>
<a NAME="E6:18_3_1_1_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E7:18_3_1_1_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:18_3_1_1_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g1</font> where <font color="Olive">g1</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">g1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> </span>}</span> 
 
;<br/>
<a NAME="E9:18_3_1_1_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">x0</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D15">PROB_1:def 15</a></i>;<br/>
<a NAME="E10:18_3_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">x0</font>
 
<b>by </b><i>, <a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E11:18_3_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">x0</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:18_3_1_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 <b>by </b><i>, , <a class="ref" href="limfunc2.html#D3">LIMFUNC2:def 3</a></i>;<br/><a NAME="E8:18_3_1_1_1"/>
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#T22">RFUNCT_2:22</a></i>;<br/><b>hence </b><a NAME="E9:18_3_1_1_1"/>
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">s</font> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 <b>by </b><i>, <a class="ref" href="limfunc1.html#T34">LIMFUNC1:34</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_3_1_1_2"><b>suppose </b></a><a NAME="E1:18_3_1_1_2"/><i><font color="Green">E74</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>   ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x0</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_3_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="E1:18_3_1_1_2_1_1"/>
(  ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>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">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> or for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_3_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:18_3_1_1_2_1_1_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>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">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</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="E3:18_3_1_1_2_1_1_1"/><i><font color="Green">E75</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="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font>
 ;<br/><a NAME="E4:18_3_1_1_2_1_1_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</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">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> )
 <b>by </b><i>, <a class="ref" href="seq_4.html#T33">SEQ_4:33</a></i>;<br/><a NAME="E5:18_3_1_1_2_1_1_1"/><i><font color="Green">E77</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:18_3_1_1_2_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:18_3_1_1_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">n</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="E3:18_3_1_1_2_1_1_1_1"/><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/>
<a NAME="E4:18_3_1_1_2_1_1_1_1"/>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/>
<a NAME="E5:18_3_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/>
<a NAME="E6:18_3_1_1_2_1_1_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E7:18_3_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:18_3_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g1</font> where <font color="Olive">g1</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g1</font> </span>}</span> 
 
;<br/>
<a NAME="E9:18_3_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font>
 
<b>by </b><i><a class="ref" href="limfunc1.html#D3">LIMFUNC1:def 3</a></i>;<br/>
<a NAME="E10:18_3_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font>
 
<b>by </b><i>, <a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E11:18_3_1_1_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:18_3_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 <b>by </b><i>, , <a class="ref" href="limfunc2.html#D6">LIMFUNC2:def 6</a></i>;<br/><a NAME="E8:18_3_1_1_2_1_1_1"/>
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#T22">RFUNCT_2:22</a></i>;<br/><b>hence </b><a NAME="E9:18_3_1_1_2_1_1_1"/>
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">s</font> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 <b>by </b><i>, <a class="ref" href="limfunc1.html#T34">LIMFUNC1:34</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_3_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:18_3_1_1_2_1_1_2"/><i><font color="Green">E81</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>   ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> )
 ;<br/><div class="add"><div><i><font color="Green">E83</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>consider </b><font color="Maroon">n</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="E2:18_3_1_1_2_1_1_2_1"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font> )
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">n</font>;<br/><b>thus </b><a NAME="E3:18_3_1_1_2_1_1_2_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E4:18_3_1_1_2_1_1_2_1"/>
<font color="Maroon">s</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">s</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/><a NAME="E5:18_3_1_1_2_1_1_2_1"/><b>then </b>
not <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:18_3_1_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:18_3_1_1_2_1_1_2_1"/>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">m1</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="E4:18_3_1_1_2_1_1_2"/><i><font color="Green">E86</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</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><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font>;<br/><a NAME="E5:18_3_1_1_2_1_1_2"/><i><font color="Green">E87</font></i>: 
 ex <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="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>]
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">M</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="E7:18_3_1_1_2_1_1_2"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">M</font>] &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="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i><div><i><font color="Green">E90</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_2"><b>now </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>consider </b><font color="Maroon">m</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="E2:18_3_1_1_2_1_1_2_2"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> )
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">m</font> = <font color="Maroon">m</font>;<br/><b>thus </b><a NAME="E3:18_3_1_1_2_1_1_2_2"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">n</font>, <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="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> holds <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">m</font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; ( 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">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">k</font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> holds <br/><font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) );<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font>];<br/><a NAME="E9:18_3_1_1_2_1_1_2"/><i><font color="Green">E94</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>3</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="18_3_1_1_2_1_1_2_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/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> );<br/>
<a NAME="E1:18_3_1_1_2_1_1_2_3"/><i><font color="Green">E97</font></i>: 
 ex <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="Maroon">S<sub>4</sub></font>[<font color="Olive">m</font>]
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">l</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="E3:18_3_1_1_2_1_1_2_3"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">l</font>] &amp; ( 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>4</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="limfunc3.html#E1"><i><font color="Green">Lemma31</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">l</font>
;<br/>

<b>thus </b><a NAME="E4:18_3_1_1_2_1_1_2_3"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Maroon">l</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">M'</font> = <font color="Maroon">M</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E12:18_3_1_1_2_1_1_2"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">M'</font> &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">S<sub>3</sub></font>[<font color="Olive">n</font>,<font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>,<font color="Maroon">F</font> <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>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S2">RECDEF_1:sch 2</a>();<br/></i><a NAME="E13:18_3_1_1_2_1_1_2"/><i><font color="Green">E100</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</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>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E14:18_3_1_1_2_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</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">F</font> = <font color="Maroon">F</font> as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/><div><i><font color="Green">E101</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_4"><b>now </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="E1:18_3_1_1_2_1_1_2_4"/>
<font color="Maroon">F</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">F</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="E2:18_3_1_1_2_1_1_2_4"/>
<font color="Maroon">F</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/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_5"><b>now </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="E1:18_3_1_1_2_1_1_2_5"/>
( <font color="Maroon">F</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>  &amp; <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:18_3_1_1_2_1_1_2_5"/>
<font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1"><i><font color="Green">Lemma31</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</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>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/><a NAME="E19:18_3_1_1_2_1_1_2"/><i><font color="Green">E103</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="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> holds <br/> ex <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="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_3_1_1_2_1_1_2_6"/><i><font color="Green">E104</font></i>: 
 ex <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="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>]
 ;<br/>

<b>consider </b><font color="Maroon">M1</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="E3:18_3_1_1_2_1_1_2_6"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">M1</font>] &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="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">M1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="limfunc3.html#E2"><i><font color="Green">Lemma35</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">M1</font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp;  ex <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="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E4:18_3_1_1_2_1_1_2_6"/><i><font color="Green">E106</font></i>: 
 ex <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="Maroon">S<sub>5</sub></font>[<font color="Olive">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_6_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">M</font>
;<br/>

<a NAME="E1:18_3_1_1_2_1_1_2_6_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:18_3_1_1_2_1_1_2_6_1"/>
<font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M1</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E1"><i><font color="Green">Lemma31</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E3:18_3_1_1_2_1_1_2_6_1"/>
<font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">M1</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>

<b>thus </b><a NAME="E4:18_3_1_1_2_1_1_2_6_1"/>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font>
 <b>by </b><i/>;<br/>

<b>take </b>
0
;<br/>

<b>thus </b><a NAME="E5:18_3_1_1_2_1_1_2_6_1"/>
<font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1"><i><font color="Green">Lemma31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:18_3_1_1_2_1_1_2_6"/><i><font color="Green">E109</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="Maroon">S<sub>5</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M1</font>
 
;<br/>
<b>consider </b><font color="Maroon">MX</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="E7:18_3_1_1_2_1_1_2_6"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">MX</font>] &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="Maroon">S<sub>5</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">MX</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(, <a class="txt" href="limfunc3.html#E3"><i><font color="Green">Lemma37</font></i></a>);<br/></i>
<a NAME="E8:18_3_1_1_2_1_1_2_6"/><i><font color="Green">E111</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">MX</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">M1</font> holds <br/><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_6_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:18_3_1_1_2_1_1_2_6_2"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">MX</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">M1</font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_6_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_6_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_3_1_1_2_1_1_2_6_2_1_1"/>
(  ex <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="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_6_2_1_1_1"><b>suppose </b></a><a NAME="E1:18_3_1_1_2_1_1_2_6_2_1_1_1"/>
 ex <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="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_3_1_1_2_1_1_2_6_2_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="limfunc3.html#E5"><i><font color="Green">Lemma41</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_6_2_1_1_2"><b>suppose </b></a><a NAME="E1:18_3_1_1_2_1_1_2_6_2_1_1_2"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_3_1_1_2_1_1_2_6_2_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="limfunc3.html#E5"><i><font color="Green">Lemma41</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="E3:18_3_1_1_2_1_1_2_6_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">m</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="E10:18_3_1_1_2_1_1_2_6"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MX</font>
 <b>by </b><i/>;<br/>
<a NAME="E11:18_3_1_1_2_1_1_2_6"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">MX</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> &amp; ( 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">MX</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">k</font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> holds <br/><font color="Maroon">F</font> <a href="seqm_3.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> <font color="Olive">k</font> ) )
 
<b>by </b><i><a class="txt" href="limfunc3.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="txt" href="limfunc3.html#E5"><i><font color="Green">Lemma41</font></i></a></i>;<br/>
<a NAME="E12:18_3_1_1_2_1_1_2_6"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">F</font> <a href="seqm_3.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> <font color="Maroon">M1</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E1"><i><font color="Green">Lemma31</font></i></a>, , , <a class="txt" href="limfunc3.html#E5"><i><font color="Green">Lemma41</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_6_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_3_1_1_2_1_1_2_6_3"/>
<font color="Maroon">F</font> <a href="seqm_3.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="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M1</font>
 ;<br/><a NAME="E2:18_3_1_1_2_1_1_2_6_3"/><b>then </b>
<font color="Maroon">F</font> <a href="seqm_3.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> <font color="Maroon">M1</font>
 <b>by </b><i><a class="ref" href="limfunc3.html#T1" target="_self">Th1</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:18_3_1_1_2_1_1_2_6_3"/>
contradiction
 <b>by </b><i><a class="txt" href="limfunc3.html#E4"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="limfunc3.html#E6"><i><font color="Green">Lemma42</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:18_3_1_1_2_1_1_2_6"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E20:18_3_1_1_2_1_1_2"/><i><font color="Green">E116</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 <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">F</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_7"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>5</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"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">F</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> <font color="Maroon">x0</font>;<br/>
<a NAME="E1:18_3_1_1_2_1_1_2_7"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[0]
 
<b>by </b><i>, <a class="txt" href="limfunc3.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>;<br/>
<a NAME="E2:18_3_1_1_2_1_1_2_7"/><i><font color="Green">E118</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>5</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_7_1"><b>proof </b></a><div class="add">

<b>let </b><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="E1:18_3_1_1_2_1_1_2_7_1"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">F</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font>
 ;<br/>

<a NAME="E2:18_3_1_1_2_1_1_2_7_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">k</font>,<font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 
<b>by </b><i><a class="txt" href="limfunc3.html#E1"><i><font color="Green">Lemma31</font></i></a></i>;<br/>
<a NAME="E3:18_3_1_1_2_1_1_2_7_1"/><b>then </b>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font>
 
;<br/>
<b>hence </b><a NAME="E4:18_3_1_1_2_1_1_2_7_1"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">F</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:18_3_1_1_2_1_1_2_7"/>
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>5</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="limfunc3.html#E3"><i><font color="Green">Lemma37</font></i></a>);</i><br/>


</div><b>end;</b></div><a NAME="E21:18_3_1_1_2_1_1_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">F</font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/><a NAME="E22:18_3_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">F</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i>, <a class="ref" href="seq_4.html#T29">SEQ_4:29</a></i>;<br/><a NAME="E23:18_3_1_1_2_1_1_2"/><i><font color="Green">E121</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 <b>by </b><i>, , <a class="ref" href="seq_4.html#T30">SEQ_4:30</a></i>;<br/><a NAME="E24:18_3_1_1_2_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#T11">RFUNCT_2:11</a></i>;<br/><a NAME="E25:18_3_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E122</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E26:18_3_1_1_2_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">x0</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:18_3_1_1_2_1_1_2_8"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">n</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="E3:18_3_1_1_2_1_1_2_8"/><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">F</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/>
<a NAME="E4:18_3_1_1_2_1_1_2_8"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">F</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E2"><i><font color="Green">Lemma35</font></i></a></i>;<br/>
<a NAME="E5:18_3_1_1_2_1_1_2_8"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g1</font> where <font color="Olive">g1</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">g1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="limfunc3.html#E5"><i><font color="Green">Lemma41</font></i></a></i>;<br/>
<a NAME="E6:18_3_1_1_2_1_1_2_8"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">x0</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D15">PROB_1:def 15</a></i>;<br/>
<a NAME="E7:18_3_1_1_2_1_1_2_8"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="txt" href="limfunc3.html#E4"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:18_3_1_1_2_1_1_2_8"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">x0</font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc3.html#E6"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E27:18_3_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">F</font></span>)</span> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 <b>by </b><i>, <a class="txt" href="limfunc3.html#E3"><i><font color="Green">Lemma37</font></i></a>, , <a class="ref" href="limfunc2.html#D3">LIMFUNC2:def 3</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font>;<br/><div><i><font color="Green">E128</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>consider </b><font color="Maroon">n</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="E2:18_3_1_1_2_1_1_2_9"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x0</font> )
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">n</font>;<br/><b>thus </b><a NAME="E3:18_3_1_1_2_1_1_2_9"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E6"><i><font color="Green">Lemma42</font></i></a></i>;<br/><a NAME="E4:18_3_1_1_2_1_1_2_9"/>
<font color="Maroon">s</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">s</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/><a NAME="E5:18_3_1_1_2_1_1_2_9"/><b>then </b>
not <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:18_3_1_1_2_1_1_2_9"/><b>then </b>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:18_3_1_1_2_1_1_2_9"/>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E6"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/></div><b>end;</b></div><a NAME="E29:18_3_1_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive">mn</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">mn</font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">mn</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font> )
 ;<br/><a NAME="E30:18_3_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E131</font></i>: 
 ex <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="Maroon">S<sub>5</sub></font>[<font color="Olive">m</font>]
 ;<br/><b>consider </b><font color="Maroon">N</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="E32:18_3_1_1_2_1_1_2"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">N</font>] &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="Maroon">S<sub>5</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">N</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="limfunc3.html#E6"><i><font color="Green">Lemma42</font></i></a>);<br/></i><div><i><font color="Green">E133</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_10"><b>now </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>consider </b><font color="Maroon">m</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="E2:18_3_1_1_2_1_1_2_10"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font> )
 <b>by </b><i><a class="txt" href="limfunc3.html#E5"><i><font color="Green">Lemma41</font></i></a></i>;<br/><b>take </b><font color="Maroon">m</font> = <font color="Maroon">m</font>;<br/><b>thus </b><a NAME="E3:18_3_1_1_2_1_1_2_10"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>6</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">n</font>, <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="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> holds <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">m</font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font> &amp; ( 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">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">k</font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font> holds <br/><font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) );<br/><b>defpred </b><font color="Maroon">S<sub>7</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">S<sub>6</sub></font>[<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font>];<br/><a NAME="E34:18_3_1_1_2_1_1_2"/><i><font color="Green">E135</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>7</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="18_3_1_1_2_1_1_2_11"><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="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>8</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font> );<br/>
<a NAME="E1:18_3_1_1_2_1_1_2_11"/><i><font color="Green">E138</font></i>: 
 ex <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="Maroon">S<sub>8</sub></font>[<font color="Olive">m</font>]
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">l</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="E3:18_3_1_1_2_1_1_2_11"/><i><font color="Green">E139</font></i>: 
( <font color="Maroon">S<sub>8</sub></font>[<font color="Maroon">l</font>] &amp; ( 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>8</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i>
<b>take </b>
<font color="Maroon">l</font>
;<br/>

<b>thus </b><a NAME="E4:18_3_1_1_2_1_1_2_11"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Maroon">l</font>]
 <b>by </b><i><a class="ref" href="limfunc3.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">N'</font> = <font color="Maroon">N</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>consider </b><font color="Maroon">G</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E37:18_3_1_1_2_1_1_2"/><i><font color="Green">E140</font></i>: 
( <font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">N'</font> &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">S<sub>7</sub></font>[<font color="Olive">n</font>,<font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>,<font color="Maroon">G</font> <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>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S2">RECDEF_1:sch 2</a>();<br/></i><a NAME="E38:18_3_1_1_2_1_1_2"/><i><font color="Green">E141</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>  &amp;  <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>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E39:18_3_1_1_2_1_1_2"/><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="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="ref" href="limfunc3.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/><div><i><font color="Green">E142</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_12"><b>now </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="E1:18_3_1_1_2_1_1_2_12"/>
<font color="Maroon">G</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">G</font>
 <b>by </b><i><a class="ref" href="limfunc3.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E2:18_3_1_1_2_1_1_2_12"/>
<font color="Maroon">G</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="ref" href="limfunc3.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_13"><b>now </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="E1:18_3_1_1_2_1_1_2_13"/>
( <font color="Maroon">G</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>  &amp; <font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:18_3_1_1_2_1_1_2_13"/>
<font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<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="limfunc3.html#T2" target="_self">Th2</a>, <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>defpred </b><font color="Maroon">S<sub>8</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font> &amp; ( for <font color="Olive">m</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="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/><a NAME="E44:18_3_1_1_2_1_1_2"/><i><font color="Green">E144</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="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font> holds <br/> ex <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="Maroon">G</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_14"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_3_1_1_2_1_1_2_14"/><i><font color="Green">E145</font></i>: 
 ex <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="Maroon">S<sub>8</sub></font>[<font color="Olive">n</font>]
 ;<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="E3:18_3_1_1_2_1_1_2_14"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">S<sub>8</sub></font>[<font color="Maroon">N1</font>] &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="Maroon">S<sub>8</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">N1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="limfunc3.html#E3:18"><i><font color="Green">E50</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>9</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">N1</font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font> &amp;  ex <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="Maroon">G</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E4:18_3_1_1_2_1_1_2_14"/><i><font color="Green">E147</font></i>: 
 ex <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="Maroon">S<sub>9</sub></font>[<font color="Olive">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_14_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">N</font>
;<br/>

<a NAME="E1:18_3_1_1_2_1_1_2_14_1"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">N</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">N1</font>
 
<b>by </b><i><a class="ref" href="limfunc3.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E2:18_3_1_1_2_1_1_2_14_1"/>
<font color="Maroon">N</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">N1</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:18_3_1_1_2_1_1_2_14_1"/>
<font color="Maroon">N</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">N1</font>
 <b>by </b><i><a class="ref" href="limfunc3.html#T3" target="_self">Th3</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>

<b>thus </b><a NAME="E4:18_3_1_1_2_1_1_2_14_1"/>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">N</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="limfunc3.html#T1" target="_self">Th1</a></i>;<br/>

<b>take </b>
0
;<br/>

<b>thus </b><a NAME="E5:18_3_1_1_2_1_1_2_14_1"/>
<font color="Maroon">G</font> <a href="seqm_3.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">N</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:18_3_1_1_2_1_1_2_14"/><i><font color="Green">E150</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="Maroon">S<sub>9</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">N1</font>
 
;<br/>
<b>consider </b><font color="Maroon">NX</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="E7:18_3_1_1_2_1_1_2_14"/><i><font color="Green">E151</font></i>: 
( <font color="Maroon">S<sub>9</sub></font>[<font color="Maroon">NX</font>] &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="Maroon">S<sub>9</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">NX</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(<a class="ref" href="limfunc3.html#T3" target="_self">Th3</a>, );<br/></i>
<a NAME="E8:18_3_1_1_2_1_1_2_14"/><i><font color="Green">E152</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">NX</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">N1</font> holds <br/><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_14_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:18_3_1_1_2_1_1_2_14_2"/><i><font color="Green">E153</font></i>: 
( <font color="Maroon">NX</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">N1</font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_14_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_14_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_3_1_1_2_1_1_2_14_2_1_1"/>
(  ex <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="Maroon">G</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or for <font color="Olive">m</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="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_14_2_1_1_1"><b>suppose </b></a><a NAME="E1:18_3_1_1_2_1_1_2_14_2_1_1_1"/>
 ex <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="Maroon">G</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_3_1_1_2_1_1_2_14_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="limfunc3.html#T4" target="_self">Th4</a>, <a class="ref" href="limfunc3.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_14_2_1_1_2"><b>suppose </b></a><a NAME="E1:18_3_1_1_2_1_1_2_14_2_1_1_2"/>
for <font color="Olive">m</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="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_3_1_1_2_1_1_2_14_2_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="limfunc3.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:18_3_1_1_2_1_1_2_14_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">m</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="E10:18_3_1_1_2_1_1_2_14"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">G</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">NX</font>
 <b>by </b><i><a class="ref" href="limfunc3.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E11:18_3_1_1_2_1_1_2_14"/><i><font color="Green">E155</font></i>: 
( <font color="Maroon">NX</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">G</font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font> &amp; ( 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">NX</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">k</font> &amp; <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font> holds <br/><font color="Maroon">G</font> <a href="seqm_3.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> <font color="Olive">k</font> ) )
 
<b>by </b><i>, <a class="ref" href="limfunc3.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E12:18_3_1_1_2_1_1_2_14"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">G</font> <a href="seqm_3.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> <font color="Maroon">N1</font>
 
<b>by </b><i>, , <a class="ref" href="limfunc3.html#T4" target="_self">Th4</a>, <a class="ref" href="limfunc3.html#T6" target="_self">Th6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_14_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_3_1_1_2_1_1_2_14_3"/>
<font color="Maroon">G</font> <a href="seqm_3.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="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">N1</font>
 ;<br/><a NAME="E2:18_3_1_1_2_1_1_2_14_3"/><b>then </b>
<font color="Maroon">G</font> <a href="seqm_3.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> <font color="Maroon">N1</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:18_3_1_1_2_1_1_2_14_3"/>
contradiction
 <b>by </b><i><a class="ref" href="limfunc3.html#T5" target="_self">Th5</a>, <a class="ref" href="limfunc3.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:18_3_1_1_2_1_1_2_14"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E45:18_3_1_1_2_1_1_2"/><i><font color="Green">E157</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 <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_15"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>9</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"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font>;<br/>
<a NAME="E1:18_3_1_1_2_1_1_2_15"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">S<sub>9</sub></font>[0]
 
<b>by </b><i><a class="ref" href="limfunc3.html#T1" target="_self">Th1</a>, , <a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>;<br/>
<a NAME="E2:18_3_1_1_2_1_1_2_15"/><i><font color="Green">E159</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>9</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>9</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_15_1"><b>proof </b></a><div class="add">

<b>let </b><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="E1:18_3_1_1_2_1_1_2_15_1"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font>
 ;<br/>

<a NAME="E2:18_3_1_1_2_1_1_2_15_1"/>
<font color="Maroon">S<sub>6</sub></font>[<font color="Maroon">G</font> <a href="seqm_3.html#K2">.</a> <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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 
<b>by </b><i/>;<br/>
<a NAME="E3:18_3_1_1_2_1_1_2_15_1"/><b>then </b>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font>
 
;<br/>
<b>hence </b><a NAME="E4:18_3_1_1_2_1_1_2_15_1"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:18_3_1_1_2_1_1_2_15"/>
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>9</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


</div><b>end;</b></div><a NAME="E46:18_3_1_1_2_1_1_2"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">G</font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/><a NAME="E47:18_3_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E161</font></i>: 
<font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">G</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i>, <a class="ref" href="seq_4.html#T29">SEQ_4:29</a></i>;<br/><a NAME="E48:18_3_1_1_2_1_1_2"/><i><font color="Green">E162</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 <b>by </b><i>, , <a class="ref" href="seq_4.html#T30">SEQ_4:30</a></i>;<br/><a NAME="E49:18_3_1_1_2_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#T11">RFUNCT_2:11</a></i>;<br/><a NAME="E50:18_3_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E163</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E51:18_3_1_1_2_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_16"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:18_3_1_1_2_1_1_2_16"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">G</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">n</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="E3:18_3_1_1_2_1_1_2_16"/><i><font color="Green">E165</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/>
<a NAME="E4:18_3_1_1_2_1_1_2_16"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E3:18"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E5:18_3_1_1_2_1_1_2_16"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g1</font> where <font color="Olive">g1</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g1</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="limfunc3.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E6:18_3_1_1_2_1_1_2_16"/><b>then </b><i><font color="Green">E167</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font>
 
<b>by </b><i><a class="ref" href="limfunc1.html#D3">LIMFUNC1:def 3</a></i>;<br/>
<a NAME="E7:18_3_1_1_2_1_1_2_16"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="limfunc3.html#T4" target="_self">Th4</a>, <a class="ref" href="limfunc3.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:18_3_1_1_2_1_1_2_16"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">x0</font></span>)</span>
 <b>by </b><i><a class="ref" href="limfunc3.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E52:18_3_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E169</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">G</font></span>)</span> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 <b>by </b><i>, , <a class="ref" href="limfunc3.html#T3" target="_self">Th3</a>, <a class="ref" href="limfunc2.html#D6">LIMFUNC2:def 6</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_17"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<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="E2:18_3_1_1_2_1_1_2_17"/><i><font color="Green">E170</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">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E4"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="limfunc1.html#D5">LIMFUNC1:def 5</a></i>;<br/><b>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="E4:18_3_1_1_2_1_1_2_17"/><i><font color="Green">E171</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">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="limfunc3.html#T5" target="_self">Th5</a>, <a class="ref" href="limfunc1.html#D5">LIMFUNC1:def 5</a></i>;<br/><b>take </b><font color="Maroon">n</font> =  <a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n2</font></span>)</span>;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E5:18_3_1_1_2_1_1_2_17"/>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/><a NAME="E6:18_3_1_1_2_1_1_2_17"/><b>then </b>
not <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E7:18_3_1_1_2_1_1_2_17"/><b>then </b><i><font color="Green">E172</font></i>: 
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>assume </b><a NAME="E8:18_3_1_1_2_1_1_2_17"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_17_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_17_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_3_1_1_2_1_1_2_17_1_1"/>
( <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> or <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</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="18_3_1_1_2_1_1_2_17_1_1_1"><b>suppose </b></a><a NAME="E1:18_3_1_1_2_1_1_2_17_1_1_1"/>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">l</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="E3:18_3_1_1_2_1_1_2_17_1_1_1"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">l</font>
 <b>by </b><i/>;<br/><a NAME="E4:18_3_1_1_2_1_1_2_17_1_1_1"/>
<font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E5:18_3_1_1_2_1_1_2_17_1_1_1"/><b>then </b><i><font color="Green">E175</font></i>: 
<font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:18_3_1_1_2_1_1_2_17_1_1_1"/><i><font color="Green">E176</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E7:18_3_1_1_2_1_1_2_17_1_1_1"/><i><font color="Green">E177</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, </i>;<br/><a NAME="E8:18_3_1_1_2_1_1_2_17_1_1_1"/>
<font color="Maroon">l</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="limfunc3.html#T8" target="_self">Th8</a>, <a class="ref" href="limfunc3.html#D1" target="_self">Def1</a>, <a class="ref" href="seqm_3.html#T7">SEQM_3:7</a></i>;<br/><a NAME="E9:18_3_1_1_2_1_1_2_17_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="limfunc3.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E10:18_3_1_1_2_1_1_2_17_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">F</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="limfunc3.html#D2" target="_self">Def2</a>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>;<br/><a NAME="E11:18_3_1_1_2_1_1_2_17_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="limfunc3.html#T8" target="_self">Th8</a>, <a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>;<br/><b>hence </b><a NAME="E12:18_3_1_1_2_1_1_2_17_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="limfunc3.html#D3" target="_self">Def3</a>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_3_1_1_2_1_1_2_17_1_1_2"><b>suppose </b></a><a NAME="E1:18_3_1_1_2_1_1_2_17_1_1_2"/>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">l</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="E3:18_3_1_1_2_1_1_2_17_1_1_2"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E2:18"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E4:18_3_1_1_2_1_1_2_17_1_1_2"/>
<font color="Maroon">G</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E5:18_3_1_1_2_1_1_2_17_1_1_2"/><b>then </b><i><font color="Green">E179</font></i>: 
<font color="Maroon">G</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:18_3_1_1_2_1_1_2_17_1_1_2"/><i><font color="Green">E181</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="limfunc3.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E7:18_3_1_1_2_1_1_2_17_1_1_2"/><i><font color="Green">E182</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, </i>;<br/><a NAME="E8:18_3_1_1_2_1_1_2_17_1_1_2"/>
<font color="Maroon">l</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="limfunc3.html#T8" target="_self">Th8</a>, <a class="ref" href="limfunc3.html#D1" target="_self">Def1</a>, <a class="ref" href="seqm_3.html#T7">SEQM_3:7</a></i>;<br/><a NAME="E9:18_3_1_1_2_1_1_2_17_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="limfunc3.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E10:18_3_1_1_2_1_1_2_17_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="limfunc3.html#D2" target="_self">Def2</a>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>;<br/><a NAME="E11:18_3_1_1_2_1_1_2_17_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="limfunc3.html#T8" target="_self">Th8</a>, <a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>;<br/><b>hence </b><a NAME="E12:18_3_1_1_2_1_1_2_17_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="limfunc3.html#D3" target="_self">Def3</a>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:18_3_1_1_2_1_1_2_17"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E54:18_3_1_1_2_1_1_2"/>
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">s</font> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 <b>by </b><i><a class="ref" href="limfunc1.html#D5">LIMFUNC1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:18_3_1_1_2"/>
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">s</font> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:18_3"/>
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">s</font> is <a href="limfunc1.html#V2">divergent_to-infty</a>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:18"/>
<font color="Maroon">f</font> <a href="limfunc3.html#R3" target="_self">is_divergent_to-infty_in</a> <font color="Maroon">x0</font>
 <b>by </b><i>, </i>;<br/>


</div>
