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

<b>let </b><font color="Maroon" title="c1">x0</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>thus </b><a NAME="E1:17"/>
( <font color="Maroon" title="c2">f</font> <a href="limfunc3.html#R2" title="LIMFUNC3:pred.2">is_divergent_to+infty_in</a> <font color="Maroon" title="c1">x0</font> implies ( <font color="Maroon" title="c2">f</font> <a href="limfunc2.html#R2" title="LIMFUNC2:pred.2">is_left_divergent_to+infty_in</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c2">f</font> <a href="limfunc2.html#R5" title="LIMFUNC2:pred.5">is_right_divergent_to+infty_in</a> <font color="Maroon" title="c1">x0</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="limfunc3.html#R2" title="LIMFUNC3:pred.2">is_divergent_to+infty_in</a> <font color="Maroon" title="c1">x0</font>
 ;<br/>

<a NAME="E2:17_1"/><b>then </b><i><font color="Green" title="E18">A2</font></i>: 
for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r2</font> holds <br/> ex <font color="Olive" title="b3">g1</font>, <font color="Olive" title="b4">g2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b3">g1</font> &amp; <font color="Olive" title="b3">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Olive" title="b3">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Olive" title="b4">g2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r2</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b4">g2</font> &amp; <font color="Olive" title="b4">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> )
 
<b>by </b><i><a class="ref" href="limfunc3.html#D2" target="_self" title="LIMFUNC3:def.2">Def2</a></i>;<br/>
<a NAME="E3:17_1"/><b>then </b><i><font color="Green" title="E19">A3</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> holds <br/> ex <font color="Olive" title="b2">g</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">g</font> &amp; <font color="Olive" title="b2">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> )
 
<b>by </b><i><a class="ref" href="limfunc3.html#T8" target="_self" title="LIMFUNC3:th.8">Th8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">s</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:17_1_1"/><i><font color="Green" title="E20">A4</font></i>: 
( <font color="Maroon" title="c3">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x0</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c1">x0</font></span>)</span> )
 ;<br/><a NAME="E2:17_1_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="limfunc3.html#T1" target="_self" title="LIMFUNC3:th.1">Th1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">s</font> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:17_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="limfunc3.html#E1:17_1_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="limfunc3.html#D2" target="_self" title="LIMFUNC3:def.2">Def2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:17_1"/>
<font color="Maroon" title="c2">f</font> <a href="limfunc2.html#R2" title="LIMFUNC2:pred.2">is_left_divergent_to+infty_in</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E3:17_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="limfunc2.html#D2" title="LIMFUNC2:def.2">LIMFUNC2:def 2</a></i>;<br/>

<a NAME="E6:17_1"/><i><font color="Green" title="E20">A5</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font> holds <br/> ex <font color="Olive" title="b2">g</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">g</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> )
 
<b>by </b><i><a class="txt" href="limfunc3.html#E2:17_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="limfunc3.html#T8" target="_self" title="LIMFUNC3:th.8">Th8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">s</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:17_1_2"/><i><font color="Green" title="E21">A6</font></i>: 
( <font color="Maroon" title="c3">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x0</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c1">x0</font></span>)</span> )
 ;<br/><a NAME="E2:17_1_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="limfunc3.html#T1" target="_self" title="LIMFUNC3:th.1">Th1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">s</font> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:17_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="limfunc3.html#E1:17_1_2"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="limfunc3.html#D2" target="_self" title="LIMFUNC3:def.2">Def2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:17_1"/>
<font color="Maroon" title="c2">f</font> <a href="limfunc2.html#R5" title="LIMFUNC2:pred.5">is_right_divergent_to+infty_in</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E6:17_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="limfunc2.html#D5" title="LIMFUNC2:def.5">LIMFUNC2:def 5</a></i>;<br/>


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

<b>assume </b><a NAME="E2:17"/><i><font color="Green" title="E17">A7</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="limfunc2.html#R2" title="LIMFUNC2:pred.2">is_left_divergent_to+infty_in</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c2">f</font> <a href="limfunc2.html#R5" title="LIMFUNC2:pred.5">is_right_divergent_to+infty_in</a> <font color="Maroon" title="c1">x0</font> )
 ;<br/>

<a NAME="E3:17"/><b>then </b><i><font color="Green" title="E18">A8</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> holds <br/> ex <font color="Olive" title="b2">g</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">g</font> &amp; <font color="Olive" title="b2">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> )
 
<b>by </b><i><a class="ref" href="limfunc2.html#D2" title="LIMFUNC2:def.2">LIMFUNC2:def 2</a></i>;<br/>
<a NAME="E4:17"/>
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font> holds <br/> ex <font color="Olive" title="b2">g</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">g</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> )
 
<b>by </b><i><a class="txt" href="limfunc3.html#E2:17"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="limfunc2.html#D5" title="LIMFUNC2:def.5">LIMFUNC2:def 5</a></i>;<br/>
<a NAME="E5:17"/><b>then </b><i><font color="Green" title="E19">A9</font></i>: 
for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r2</font> holds <br/> ex <font color="Olive" title="b3">g1</font>, <font color="Olive" title="b4">g2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b3">g1</font> &amp; <font color="Olive" title="b3">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Olive" title="b3">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Olive" title="b4">g2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r2</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b4">g2</font> &amp; <font color="Olive" title="b4">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> )
 
<b>by </b><i><a class="txt" href="limfunc3.html#E3:17"><i><font color="Green" title="E18">A8</font></i></a>, <a class="ref" href="limfunc3.html#T8" target="_self" title="LIMFUNC3:th.8">Th8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">s</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:17_2"/><i><font color="Green" title="E20">A10</font></i>: 
( <font color="Maroon" title="c3">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x0</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_2_1_1"/>
(  ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> holds <br/><font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> or for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">x0</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_1"><b>suppose </b></a><a NAME="E1:17_2_1_1_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> holds <br/><font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:17_2_1_1_1"/><i><font color="Green" title="E21">A11</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font>
 ;<br/><a NAME="E4:17_2_1_1_1"/><i><font color="Green" title="E22">A12</font></i>: 
( <font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x0</font> )
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:17_2"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="seq_4.html#T33" title="SEQ_4:th.33">SEQ_4:33</a></i>;<br/><a NAME="E5:17_2_1_1_1"/><i><font color="Green" title="E23">A13</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:17_2"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E6:17_2_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c1">x0</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_2_1_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:17_2_1_1_1_1"/><i><font color="Green" title="E24">A14</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9" title="RFUNCT_2:th.9">RFUNCT_2:9</a></i>;<br/>
<a NAME="E4:17_2_1_1_1_1"/>
<font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">s</font>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T10" title="RFUNCT_2:th.10">RFUNCT_2:10</a></i>;<br/>
<a NAME="E5:17_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E25">A15</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">s</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E3:17_2_1_1_1_1"><i><font color="Green" title="E24">A14</font></i></a>, <a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>;<br/>
<a NAME="E6:17_2_1_1_1_1"/>
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">k</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E7:17_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E3:17_2_1_1_1"><i><font color="Green" title="E21">A11</font></i></a></i>;<br/>
<a NAME="E8:17_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g1</font> where <font color="Olive" title="b1">g1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b1">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> </span>}</span> 
 
;<br/>
<a NAME="E9:17_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c1">x0</font>
 
<b>by </b><i><a class="ref" href="limfunc1.html#T151" title="LIMFUNC1:th.151">LIMFUNC1:151</a></i>;<br/>
<a NAME="E10:17_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c1">x0</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E3:17_2_1_1_1_1"><i><font color="Green" title="E24">A14</font></i></a>, <a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E11:17_2_1_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c1">x0</font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc3.html#E5:17_2_1_1_1"><i><font color="Green" title="E23">A13</font></i></a>, <a class="txt" href="limfunc3.html#E5:17_2_1_1_1_1"><i><font color="Green" title="E25">A15</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:17_2_1_1_1"/><b>then </b><i><font color="Green" title="E24">A16</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc3.html#E2:17"><i><font color="Green" title="E17">A7</font></i></a>, <a class="txt" href="limfunc3.html#E4:17_2_1_1_1"><i><font color="Green" title="E22">A12</font></i></a>, <a class="ref" href="limfunc2.html#D2" title="LIMFUNC2:def.2">LIMFUNC2:def 2</a></i>;<br/><a NAME="E8:17_2_1_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E5:17_2_1_1_1"><i><font color="Green" title="E23">A13</font></i></a>, <a class="ref" href="rfunct_2.html#T22" title="RFUNCT_2:th.22">RFUNCT_2:22</a></i>;<br/><b>hence </b><a NAME="E9:17_2_1_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">s</font> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc3.html#E7:17_2_1_1_1"><i><font color="Green" title="E24">A16</font></i></a>, <a class="ref" href="limfunc1.html#T34" title="LIMFUNC1:th.34">LIMFUNC1:34</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_2"><b>suppose </b></a><a NAME="E1:17_2_1_1_2"/><i><font color="Green" title="E21">A17</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">x0</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_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="E1:17_2_1_1_2_1_1"/>
(  ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> holds <br/><font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">n</font> or for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">n</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:17_2_1_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> holds <br/><font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">n</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:17_2_1_1_2_1_1_1"/><i><font color="Green" title="E22">A18</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font>
 ;<br/><a NAME="E4:17_2_1_1_2_1_1_1"/><i><font color="Green" title="E23">A19</font></i>: 
( <font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x0</font> )
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:17_2"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="seq_4.html#T33" title="SEQ_4:th.33">SEQ_4:33</a></i>;<br/><a NAME="E5:17_2_1_1_2_1_1_1"/><i><font color="Green" title="E24">A20</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:17_2"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E6:17_2_1_1_2_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c1">x0</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_2_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:17_2_1_1_2_1_1_1_1"/><i><font color="Green" title="E25">A21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9" title="RFUNCT_2:th.9">RFUNCT_2:9</a></i>;<br/>
<a NAME="E4:17_2_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">s</font>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T10" title="RFUNCT_2:th.10">RFUNCT_2:10</a></i>;<br/>
<a NAME="E5:17_2_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E26">A22</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">s</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E3:17_2_1_1_2_1_1_1_1"><i><font color="Green" title="E25">A21</font></i></a>, <a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>;<br/>
<a NAME="E6:17_2_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">k</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E7:17_2_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E3:17_2_1_1_2_1_1_1"><i><font color="Green" title="E22">A18</font></i></a></i>;<br/>
<a NAME="E8:17_2_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g1</font> where <font color="Olive" title="b1">g1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g1</font> </span>}</span> 
 
;<br/>
<a NAME="E9:17_2_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c1">x0</font>
 
<b>by </b><i><a class="ref" href="limfunc1.html#T152" title="LIMFUNC1:th.152">LIMFUNC1:152</a></i>;<br/>
<a NAME="E10:17_2_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c1">x0</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E3:17_2_1_1_2_1_1_1_1"><i><font color="Green" title="E25">A21</font></i></a>, <a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E11:17_2_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c1">x0</font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc3.html#E5:17_2_1_1_2_1_1_1"><i><font color="Green" title="E24">A20</font></i></a>, <a class="txt" href="limfunc3.html#E5:17_2_1_1_2_1_1_1_1"><i><font color="Green" title="E26">A22</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:17_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E25">A23</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc3.html#E2:17"><i><font color="Green" title="E17">A7</font></i></a>, <a class="txt" href="limfunc3.html#E4:17_2_1_1_2_1_1_1"><i><font color="Green" title="E23">A19</font></i></a>, <a class="ref" href="limfunc2.html#D5" title="LIMFUNC2:def.5">LIMFUNC2:def 5</a></i>;<br/><a NAME="E8:17_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E5:17_2_1_1_2_1_1_1"><i><font color="Green" title="E24">A20</font></i></a>, <a class="ref" href="rfunct_2.html#T22" title="RFUNCT_2:th.22">RFUNCT_2:22</a></i>;<br/><b>hence </b><a NAME="E9:17_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">s</font> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc3.html#E7:17_2_1_1_2_1_1_1"><i><font color="Green" title="E25">A23</font></i></a>, <a class="ref" href="limfunc1.html#T34" title="LIMFUNC1:th.34">LIMFUNC1:34</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:17_2_1_1_2_1_1_2"/><i><font color="Green" title="E22">A24</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">n</font> )
 ;<br/><div class="add"><div><i><font color="Green" title="E23">A25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>consider </b><font color="Maroon" title="c5">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E2:17_2_1_1_2_1_1_2_1"/><i><font color="Green" title="E24">A26</font></i>: 
( <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x0</font> )
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:17_2_1_1_2_1_1_2"><i><font color="Green" title="E22">A24</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c6">n</font> = <font color="Maroon" title="c5">n</font>;<br/><b>thus </b><a NAME="E3:17_2_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E2:17_2_1_1_2_1_1_2_1"><i><font color="Green" title="E24">A26</font></i></a></i>;<br/><a NAME="E4:17_2_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10" title="RFUNCT_2:th.10">RFUNCT_2:10</a></i>;<br/><a NAME="E5:17_2_1_1_2_1_1_2_1"/><b>then </b>
not <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:17_2"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:17_2_1_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:17_2_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E2:17_2_1_1_2_1_1_2_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c4">m1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:17_2_1_1_2_1_1_2"/><i><font color="Green" title="E24">A27</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m1</font> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> )
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font>;<br/><a NAME="E5:17_2_1_1_2_1_1_2"/><i><font color="Green" title="E25">A28</font></i>: 
 ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>]
 <b>by </b><i><a class="txt" href="limfunc3.html#E4:17_2_1_1_2_1_1_2"><i><font color="Green" title="E24">A27</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c5">M</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E7:17_2_1_1_2_1_1_2"/><i><font color="Green" title="E26">A29</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">M</font>] &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c5">M</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="limfunc3.html#E5:17_2_1_1_2_1_1_2"><i><font color="Green" title="E25">A28</font></i></a>);<br/></i><div><i><font color="Green" title="E27">A30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>consider </b><font color="Maroon" title="c7">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E2:17_2_1_1_2_1_1_2_2"/><i><font color="Green" title="E28">A31</font></i>: 
( <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> )
 <b>by </b><i><a class="txt" href="limfunc3.html#E2:17_2_1_1_2_1_1_2"><i><font color="Green" title="E23">A25</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c8">m</font> = <font color="Maroon" title="c7">m</font>;<br/><b>thus </b><a NAME="E3:17_2_1_1_2_1_1_2_2"/>
( <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">m</font> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> )
 <b>by </b><i><a class="txt" href="limfunc3.html#E2:17_2_1_1_2_1_1_2_2"><i><font color="Green" title="E28">A31</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> holds <br/>( <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; ( for <font color="Olive" title="b3">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b3">k</font> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> holds <br/><font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">k</font> ) );<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><font color="Maroon">S<sub>2</sub></font>[$2,$3];<br/><a NAME="E9:17_2_1_1_2_1_1_2"/><i><font color="Green" title="E28">A32</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">n</font>, <font color="Maroon" title="c7">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> $1 &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> );<br/>
<a NAME="E1:17_2_1_1_2_1_1_2_3"/><i><font color="Green" title="E29">A33</font></i>: 
 ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">m</font>]
 
<b>by </b><i><a class="txt" href="limfunc3.html#E8:17_2_1_1_2_1_1_2"><i><font color="Green" title="E27">A30</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">l</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:17_2_1_1_2_1_1_2_3"/><i><font color="Green" title="E30">A34</font></i>: 
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c8">l</font>] &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon" title="c8">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="limfunc3.html#E1:17_2_1_1_2_1_1_2_3"><i><font color="Green" title="E29">A33</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c8">l</font>
;<br/>

<b>thus </b><a NAME="E4:17_2_1_1_2_1_1_2_3"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">l</font>]
 <b>by </b><i><a class="txt" href="limfunc3.html#E3:17_2_1_1_2_1_1_2_3"><i><font color="Green" title="E30">A34</font></i></a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c6">M'</font> = <font color="Maroon" title="c5">M</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><b>consider </b><font color="Maroon" title="c7">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E12:17_2_1_1_2_1_1_2"/><i><font color="Green" title="E29">A35</font></i>: 
( <font color="Maroon" title="c7">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M'</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c7">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c7">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S2" title="RECDEF_1:sch.2">RECDEF_1:sch 2</a>(<a class="txt" href="limfunc3.html#E9:17_2_1_1_2_1_1_2"><i><font color="Green" title="E28">A32</font></i></a>);<br/></i><a NAME="E13:17_2_1_1_2_1_1_2"/><i><font color="Green" title="E30">A36</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><a NAME="E14:17_2_1_1_2_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">F</font> = <font color="Maroon" title="c7">F</font> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> <b>by </b><i><a class="txt" href="limfunc3.html#E13:17_2_1_1_2_1_1_2"><i><font color="Green" title="E30">A36</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/><div><i><font color="Green" title="E31">A37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:17_2_1_1_2_1_1_2_4"/>
<font color="Maroon" title="c8">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">F</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E13:17_2_1_1_2_1_1_2"><i><font color="Green" title="E30">A36</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E2:17_2_1_1_2_1_1_2_4"/>
<font color="Maroon" title="c8">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="limfunc3.html#E13:17_2_1_1_2_1_1_2"><i><font color="Green" title="E30">A36</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:17_2_1_1_2_1_1_2_5"/>
( <font color="Maroon" title="c8">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="txt" href="limfunc3.html#E16:17_2_1_1_2_1_1_2"><i><font color="Green" title="E31">A37</font></i></a></i>;<br/><b>hence </b><a NAME="E2:17_2_1_1_2_1_1_2_5"/>
<font color="Maroon" title="c8">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="limfunc3.html#E12:17_2_1_1_2_1_1_2"><i><font color="Green" title="E29">A35</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c9">F</font> = <font color="Maroon" title="c8">F</font> as  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">increasing</a> <a href="seqm_3.html#NM1" title="SEQM_3:NM.1">Seq_of_Nat</a> <b>by </b><i><a class="txt" href="limfunc3.html#E13:17_2_1_1_2_1_1_2"><i><font color="Green" title="E30">A36</font></i></a>, <a class="ref" href="seqm_3.html#D8" title="SEQM_3:def.8">SEQM_3:def 8</a>, <a class="ref" href="seqm_3.html#D11" title="SEQM_3:def.11">SEQM_3:def 11</a></i>;<br/><a NAME="E19:17_2_1_1_2_1_1_2"/><i><font color="Green" title="E32">A38</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> holds <br/> ex <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_6"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; ( for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> $1 ) );<br/>
<b>assume </b><a NAME="E1:17_2_1_1_2_1_1_2_6"/><i><font color="Green" title="E33">A39</font></i>: 
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>]
 ;<br/>

<b>consider </b><font color="Maroon" title="c10">M1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:17_2_1_1_2_1_1_2_6"/><i><font color="Green" title="E34">A40</font></i>: 
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c10">M1</font>] &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c10">M1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="limfunc3.html#E1:17_2_1_1_2_1_1_2_6"><i><font color="Green" title="E33">A39</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">M1</font> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp;  ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 );<br/>
<a NAME="E4:17_2_1_1_2_1_1_2_6"/><i><font color="Green" title="E35">A41</font></i>: 
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_6_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c5">M</font>
;<br/>

<a NAME="E1:17_2_1_1_2_1_1_2_6_1"/><i><font color="Green" title="E36">A42</font></i>: 
<font color="Maroon" title="c5">M</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">M1</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E7:17_2_1_1_2_1_1_2"><i><font color="Green" title="E26">A29</font></i></a>, <a class="txt" href="limfunc3.html#E3:17_2_1_1_2_1_1_2_6"><i><font color="Green" title="E34">A40</font></i></a></i>;<br/>
<a NAME="E2:17_2_1_1_2_1_1_2_6_1"/>
<font color="Maroon" title="c5">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">M1</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E12:17_2_1_1_2_1_1_2"><i><font color="Green" title="E29">A35</font></i></a>, <a class="txt" href="limfunc3.html#E3:17_2_1_1_2_1_1_2_6"><i><font color="Green" title="E34">A40</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:17_2_1_1_2_1_1_2_6_1"/>
<font color="Maroon" title="c5">M</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">M1</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:17_2_1_1_2_1_1_2_6_1"><i><font color="Green" title="E36">A42</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>

<b>thus </b><a NAME="E4:17_2_1_1_2_1_1_2_6_1"/>
<font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">M</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E7:17_2_1_1_2_1_1_2"><i><font color="Green" title="E26">A29</font></i></a></i>;<br/>

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

<b>thus </b><a NAME="E5:17_2_1_1_2_1_1_2_6_1"/>
<font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E12:17_2_1_1_2_1_1_2"><i><font color="Green" title="E29">A35</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:17_2_1_1_2_1_1_2_6"/><i><font color="Green" title="E36">A43</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">M1</font>
 
;<br/>
<b>consider </b><font color="Maroon" title="c11">MX</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E7:17_2_1_1_2_1_1_2_6"/><i><font color="Green" title="E37">A44</font></i>: 
( <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c11">MX</font>] &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">MX</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="limfunc3.html#E5:17_2_1_1_2_1_1_2_6"><i><font color="Green" title="E36">A43</font></i></a>, <a class="txt" href="limfunc3.html#E4:17_2_1_1_2_1_1_2_6"><i><font color="Green" title="E35">A41</font></i></a>);<br/></i>
<a NAME="E8:17_2_1_1_2_1_1_2_6"/><i><font color="Green" title="E38">A45</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c11">MX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">M1</font> holds <br/><font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">x0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_6_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c12">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><a NAME="E2:17_2_1_1_2_1_1_2_6_2"/><i><font color="Green" title="E39">A46</font></i>: 
( <font color="Maroon" title="c11">MX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">k</font> &amp; <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">M1</font> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_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="17_2_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="E1:17_2_1_1_2_1_1_2_6_2_1_1"/>
(  ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">k</font> or for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_6_2_1_1_1"><b>suppose </b></a><a NAME="E1:17_2_1_1_2_1_1_2_6_2_1_1_1"/>
 ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_2_1_1_2_1_1_2_6_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="limfunc3.html#E7:17_2_1_1_2_1_1_2_6"><i><font color="Green" title="E37">A44</font></i></a>, <a class="txt" href="limfunc3.html#E2:17_2_1_1_2_1_1_2_6_2"><i><font color="Green" title="E39">A46</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_6_2_1_1_2"><b>suppose </b></a><a NAME="E1:17_2_1_1_2_1_1_2_6_2_1_1_2"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_2_1_1_2_1_1_2_6_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="limfunc3.html#E3:17_2_1_1_2_1_1_2_6"><i><font color="Green" title="E34">A40</font></i></a>, <a class="txt" href="limfunc3.html#E2:17_2_1_1_2_1_1_2_6_2"><i><font color="Green" title="E39">A46</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:17_2_1_1_2_1_1_2_6_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c12">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E10:17_2_1_1_2_1_1_2_6"/><i><font color="Green" title="E39">A47</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">MX</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E7:17_2_1_1_2_1_1_2_6"><i><font color="Green" title="E37">A44</font></i></a></i>;<br/>
<a NAME="E11:17_2_1_1_2_1_1_2_6"/><i><font color="Green" title="E40">A48</font></i>: 
( <font color="Maroon" title="c11">MX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c11">MX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> holds <br/><font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> ) )
 
<b>by </b><i><a class="txt" href="limfunc3.html#E12:17_2_1_1_2_1_1_2"><i><font color="Green" title="E29">A35</font></i></a>, <a class="txt" href="limfunc3.html#E10:17_2_1_1_2_1_1_2_6"><i><font color="Green" title="E39">A47</font></i></a></i>;<br/>
<a NAME="E12:17_2_1_1_2_1_1_2_6"/><i><font color="Green" title="E41">A49</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">M1</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E12:17_2_1_1_2_1_1_2"><i><font color="Green" title="E29">A35</font></i></a>, <a class="txt" href="limfunc3.html#E3:17_2_1_1_2_1_1_2_6"><i><font color="Green" title="E34">A40</font></i></a>, <a class="txt" href="limfunc3.html#E7:17_2_1_1_2_1_1_2_6"><i><font color="Green" title="E37">A44</font></i></a>, <a class="txt" href="limfunc3.html#E10:17_2_1_1_2_1_1_2_6"><i><font color="Green" title="E39">A47</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_6_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_2_1_1_2_1_1_2_6_3"/>
<font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">M1</font>
 ;<br/><a NAME="E2:17_2_1_1_2_1_1_2_6_3"/><b>then </b>
<font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">M1</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E12:17_2_1_1_2_1_1_2_6"><i><font color="Green" title="E41">A49</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:17_2_1_1_2_1_1_2_6_3"/>
contradiction
 <b>by </b><i><a class="txt" href="limfunc3.html#E8:17_2_1_1_2_1_1_2_6"><i><font color="Green" title="E38">A45</font></i></a>, <a class="txt" href="limfunc3.html#E11:17_2_1_1_2_1_1_2_6"><i><font color="Green" title="E40">A48</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:17_2_1_1_2_1_1_2_6"/>
contradiction
 <b>by </b><i><a class="txt" href="limfunc3.html#E3:17_2_1_1_2_1_1_2_6"><i><font color="Green" title="E34">A40</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E20:17_2_1_1_2_1_1_2"/><i><font color="Green" title="E33">A50</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_7"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font>;<br/>
<a NAME="E1:17_2_1_1_2_1_1_2_7"/><i><font color="Green" title="E34">A51</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 
<b>by </b><i><a class="txt" href="limfunc3.html#E7:17_2_1_1_2_1_1_2"><i><font color="Green" title="E26">A29</font></i></a>, <a class="txt" href="limfunc3.html#E12:17_2_1_1_2_1_1_2"><i><font color="Green" title="E29">A35</font></i></a>, <a class="ref" href="seqm_3.html#T31" title="SEQM_3:th.31">SEQM_3:31</a></i>;<br/>
<a NAME="E2:17_2_1_1_2_1_1_2_7"/><i><font color="Green" title="E35">A52</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:17_2_1_1_2_1_1_2_7_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font>
 ;<br/>

<a NAME="E2:17_2_1_1_2_1_1_2_7_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>]
 
<b>by </b><i><a class="txt" href="limfunc3.html#E12:17_2_1_1_2_1_1_2"><i><font color="Green" title="E29">A35</font></i></a></i>;<br/>
<a NAME="E3:17_2_1_1_2_1_1_2_7_1"/><b>then </b>
<font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font>
 
;<br/>
<b>hence </b><a NAME="E4:17_2_1_1_2_1_1_2_7_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#T31" title="SEQM_3:th.31">SEQM_3:31</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:17_2_1_1_2_1_1_2_7"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="limfunc3.html#E1:17_2_1_1_2_1_1_2_7"><i><font color="Green" title="E34">A51</font></i></a>, <a class="txt" href="limfunc3.html#E2:17_2_1_1_2_1_1_2_7"><i><font color="Green" title="E35">A52</font></i></a>);</i><br/>


</div><b>end;</b></div><a NAME="E21:17_2_1_1_2_1_1_2"/><i><font color="Green" title="E34">A53</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c9">F</font> is    <a href="seqm_3.html#M1" title="SEQM_3:mode.1">subsequence</a> of <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D10" title="SEQM_3:def.10">SEQM_3:def 10</a></i>;<br/><a NAME="E22:17_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E35">A54</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c9">F</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:17_2"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="seq_4.html#T29" title="SEQ_4:th.29">SEQ_4:29</a></i>;<br/><a NAME="E23:17_2_1_1_2_1_1_2"/><i><font color="Green" title="E36">A55</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:17_2"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="limfunc3.html#E21:17_2_1_1_2_1_1_2"><i><font color="Green" title="E34">A53</font></i></a>, <a class="ref" href="seq_4.html#T30" title="SEQ_4:th.30">SEQ_4:30</a></i>;<br/><a NAME="E24:17_2_1_1_2_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E21:17_2_1_1_2_1_1_2"><i><font color="Green" title="E34">A53</font></i></a>, <a class="ref" href="rfunct_2.html#T11" title="RFUNCT_2:th.11">RFUNCT_2:11</a></i>;<br/><a NAME="E25:17_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E37">A56</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:17_2"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E26:17_2_1_1_2_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c1">x0</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_2_1_1_2_1_1_2_8"/><i><font color="Green" title="E38">A57</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c9">F</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:17_2_1_1_2_1_1_2_8"/><i><font color="Green" title="E39">A58</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9" title="RFUNCT_2:th.9">RFUNCT_2:9</a></i>;<br/>
<a NAME="E4:17_2_1_1_2_1_1_2_8"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E20:17_2_1_1_2_1_1_2"><i><font color="Green" title="E33">A50</font></i></a></i>;<br/>
<a NAME="E5:17_2_1_1_2_1_1_2_8"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g1</font> where <font color="Olive" title="b1">g1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b1">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="limfunc3.html#E3:17_2_1_1_2_1_1_2_8"><i><font color="Green" title="E39">A58</font></i></a></i>;<br/>
<a NAME="E6:17_2_1_1_2_1_1_2_8"/><b>then </b><i><font color="Green" title="E40">A59</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c1">x0</font>
 
<b>by </b><i><a class="ref" href="limfunc1.html#T151" title="LIMFUNC1:th.151">LIMFUNC1:151</a></i>;<br/>
<a NAME="E7:17_2_1_1_2_1_1_2_8"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E25:17_2_1_1_2_1_1_2"><i><font color="Green" title="E37">A56</font></i></a>, <a class="txt" href="limfunc3.html#E1:17_2_1_1_2_1_1_2_8"><i><font color="Green" title="E38">A57</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:17_2_1_1_2_1_1_2_8"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c1">x0</font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc3.html#E6:17_2_1_1_2_1_1_2_8"><i><font color="Green" title="E40">A59</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E27:17_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E38">A60</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c9">F</font></span>)</span> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc3.html#E2:17"><i><font color="Green" title="E17">A7</font></i></a>, <a class="txt" href="limfunc3.html#E22:17_2_1_1_2_1_1_2"><i><font color="Green" title="E35">A54</font></i></a>, <a class="txt" href="limfunc3.html#E23:17_2_1_1_2_1_1_2"><i><font color="Green" title="E36">A55</font></i></a>, <a class="ref" href="limfunc2.html#D2" title="LIMFUNC2:def.2">LIMFUNC2:def 2</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font>;<br/><div><i><font color="Green" title="E39">A61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>consider </b><font color="Maroon" title="c11">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E2:17_2_1_1_2_1_1_2_9"/><i><font color="Green" title="E40">A62</font></i>: 
( <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">x0</font> )
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:17_2_1_1_2"><i><font color="Green" title="E21">A17</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c12">n</font> = <font color="Maroon" title="c11">n</font>;<br/><b>thus </b><a NAME="E3:17_2_1_1_2_1_1_2_9"/>
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E2:17_2_1_1_2_1_1_2_9"><i><font color="Green" title="E40">A62</font></i></a></i>;<br/><a NAME="E4:17_2_1_1_2_1_1_2_9"/>
<font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10" title="RFUNCT_2:th.10">RFUNCT_2:10</a></i>;<br/><a NAME="E5:17_2_1_1_2_1_1_2_9"/><b>then </b>
not <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:17_2"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:17_2_1_1_2_1_1_2_9"/><b>then </b>
<font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:17_2_1_1_2_1_1_2_9"/>
<font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E2:17_2_1_1_2_1_1_2_9"><i><font color="Green" title="E40">A62</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/></div><b>end;</b></div><a NAME="E29:17_2_1_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">mn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">mn</font> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">mn</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font> )
 ;<br/><a NAME="E30:17_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E40">A63</font></i>: 
 ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">m</font>]
 ;<br/><b>consider </b><font color="Maroon" title="c10">N</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E32:17_2_1_1_2_1_1_2"/><i><font color="Green" title="E41">A64</font></i>: 
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c10">N</font>] &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c10">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="limfunc3.html#E30:17_2_1_1_2_1_1_2"><i><font color="Green" title="E40">A63</font></i></a>);<br/></i><div><i><font color="Green" title="E42">A65</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>consider </b><font color="Maroon" title="c12">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E2:17_2_1_1_2_1_1_2_10"/><i><font color="Green" title="E43">A66</font></i>: 
( <font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font> )
 <b>by </b><i><a class="txt" href="limfunc3.html#E28:17_2_1_1_2_1_1_2"><i><font color="Green" title="E39">A61</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c13">m</font> = <font color="Maroon" title="c12">m</font>;<br/><b>thus </b><a NAME="E3:17_2_1_1_2_1_1_2_10"/>
( <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c13">m</font> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font> )
 <b>by </b><i><a class="txt" href="limfunc3.html#E2:17_2_1_1_2_1_1_2_10"><i><font color="Green" title="E43">A66</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> holds <br/>( <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font> &amp; ( for <font color="Olive" title="b3">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b3">k</font> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font> holds <br/><font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">k</font> ) );<br/><b>defpred </b><font color="Maroon">S<sub>6</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><font color="Maroon">S<sub>5</sub></font>[$2,$3];<br/><a NAME="E34:17_2_1_1_2_1_1_2"/><i><font color="Green" title="E43">A67</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">n</font>, <font color="Maroon" title="c12">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>7</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> $1 &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font> );<br/>
<a NAME="E1:17_2_1_1_2_1_1_2_11"/><i><font color="Green" title="E44">A68</font></i>: 
 ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive" title="b1">m</font>]
 
<b>by </b><i><a class="txt" href="limfunc3.html#E33:17_2_1_1_2_1_1_2"><i><font color="Green" title="E42">A65</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">l</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:17_2_1_1_2_1_1_2_11"/><i><font color="Green" title="E45">A69</font></i>: 
( <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c13">l</font>] &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon" title="c13">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="limfunc3.html#E1:17_2_1_1_2_1_1_2_11"><i><font color="Green" title="E44">A68</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c13">l</font>
;<br/>

<b>thus </b><a NAME="E4:17_2_1_1_2_1_1_2_11"/>
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">l</font>]
 <b>by </b><i><a class="txt" href="limfunc3.html#E3:17_2_1_1_2_1_1_2_11"><i><font color="Green" title="E45">A69</font></i></a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c11">N'</font> = <font color="Maroon" title="c10">N</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><b>consider </b><font color="Maroon" title="c12">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E37:17_2_1_1_2_1_1_2"/><i><font color="Green" title="E44">A70</font></i>: 
( <font color="Maroon" title="c12">G</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">N'</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c12">G</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c12">G</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S2" title="RECDEF_1:sch.2">RECDEF_1:sch 2</a>(<a class="txt" href="limfunc3.html#E34:17_2_1_1_2_1_1_2"><i><font color="Green" title="E43">A67</font></i></a>);<br/></i><a NAME="E38:17_2_1_1_2_1_1_2"/><i><font color="Green" title="E45">A71</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><a NAME="E39:17_2_1_1_2_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">G</font> = <font color="Maroon" title="c12">G</font> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> <b>by </b><i><a class="txt" href="limfunc3.html#E38:17_2_1_1_2_1_1_2"><i><font color="Green" title="E45">A71</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/><div><i><font color="Green" title="E46">A72</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_12"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:17_2_1_1_2_1_1_2_12"/>
<font color="Maroon" title="c13">G</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">G</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E38:17_2_1_1_2_1_1_2"><i><font color="Green" title="E45">A71</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E2:17_2_1_1_2_1_1_2_12"/>
<font color="Maroon" title="c13">G</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="limfunc3.html#E38:17_2_1_1_2_1_1_2"><i><font color="Green" title="E45">A71</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_13"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:17_2_1_1_2_1_1_2_13"/>
( <font color="Maroon" title="c13">G</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c13">G</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="txt" href="limfunc3.html#E41:17_2_1_1_2_1_1_2"><i><font color="Green" title="E46">A72</font></i></a></i>;<br/><b>hence </b><a NAME="E2:17_2_1_1_2_1_1_2_13"/>
<font color="Maroon" title="c13">G</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c13">G</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="limfunc3.html#E37:17_2_1_1_2_1_1_2"><i><font color="Green" title="E44">A70</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c14">G</font> = <font color="Maroon" title="c13">G</font> as  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">increasing</a> <a href="seqm_3.html#NM1" title="SEQM_3:NM.1">Seq_of_Nat</a> <b>by </b><i><a class="txt" href="limfunc3.html#E38:17_2_1_1_2_1_1_2"><i><font color="Green" title="E45">A71</font></i></a>, <a class="ref" href="seqm_3.html#D8" title="SEQM_3:def.8">SEQM_3:def 8</a>, <a class="ref" href="seqm_3.html#D11" title="SEQM_3:def.11">SEQM_3:def 11</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>7</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font> &amp; ( for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c14">G</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> $1 ) );<br/><a NAME="E44:17_2_1_1_2_1_1_2"/><i><font color="Green" title="E47">A73</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font> holds <br/> ex <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c14">G</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_14"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_2_1_1_2_1_1_2_14"/><i><font color="Green" title="E48">A74</font></i>: 
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive" title="b1">n</font>]
 ;<br/>

<b>consider </b><font color="Maroon" title="c15">N1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:17_2_1_1_2_1_1_2_14"/><i><font color="Green" title="E49">A75</font></i>: 
( <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c15">N1</font>] &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c15">N1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="limfunc3.html#E1:17_2_1_1_2_1_1_2_14"><i><font color="Green" title="E48">A74</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>8</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">N1</font> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font> &amp;  ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c14">G</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 );<br/>
<a NAME="E4:17_2_1_1_2_1_1_2_14"/><i><font color="Green" title="E50">A76</font></i>: 
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>8</sub></font>[<font color="Olive" title="b1">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_14_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c10">N</font>
;<br/>

<a NAME="E1:17_2_1_1_2_1_1_2_14_1"/><i><font color="Green" title="E51">A77</font></i>: 
<font color="Maroon" title="c10">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">N1</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E32:17_2_1_1_2_1_1_2"><i><font color="Green" title="E41">A64</font></i></a>, <a class="txt" href="limfunc3.html#E3:17_2_1_1_2_1_1_2_14"><i><font color="Green" title="E49">A75</font></i></a></i>;<br/>
<a NAME="E2:17_2_1_1_2_1_1_2_14_1"/>
<font color="Maroon" title="c10">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">N1</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E37:17_2_1_1_2_1_1_2"><i><font color="Green" title="E44">A70</font></i></a>, <a class="txt" href="limfunc3.html#E3:17_2_1_1_2_1_1_2_14"><i><font color="Green" title="E49">A75</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:17_2_1_1_2_1_1_2_14_1"/>
<font color="Maroon" title="c10">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">N1</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:17_2_1_1_2_1_1_2_14_1"><i><font color="Green" title="E51">A77</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>

<b>thus </b><a NAME="E4:17_2_1_1_2_1_1_2_14_1"/>
<font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">N</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E32:17_2_1_1_2_1_1_2"><i><font color="Green" title="E41">A64</font></i></a></i>;<br/>

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

<b>thus </b><a NAME="E5:17_2_1_1_2_1_1_2_14_1"/>
<font color="Maroon" title="c14">G</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">N</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E37:17_2_1_1_2_1_1_2"><i><font color="Green" title="E44">A70</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:17_2_1_1_2_1_1_2_14"/><i><font color="Green" title="E51">A78</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>8</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">N1</font>
 
;<br/>
<b>consider </b><font color="Maroon" title="c16">NX</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E7:17_2_1_1_2_1_1_2_14"/><i><font color="Green" title="E52">A79</font></i>: 
( <font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c16">NX</font>] &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>8</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">NX</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="limfunc3.html#E5:17_2_1_1_2_1_1_2_14"><i><font color="Green" title="E51">A78</font></i></a>, <a class="txt" href="limfunc3.html#E4:17_2_1_1_2_1_1_2_14"><i><font color="Green" title="E50">A76</font></i></a>);<br/></i>
<a NAME="E8:17_2_1_1_2_1_1_2_14"/><i><font color="Green" title="E53">A80</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c16">NX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">N1</font> holds <br/><font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_14_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c17">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><a NAME="E2:17_2_1_1_2_1_1_2_14_2"/><i><font color="Green" title="E54">A81</font></i>: 
( <font color="Maroon" title="c16">NX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">k</font> &amp; <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">N1</font> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_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="17_2_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="E1:17_2_1_1_2_1_1_2_14_2_1_1"/>
(  ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c14">G</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">k</font> or for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c14">G</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_14_2_1_1_1"><b>suppose </b></a><a NAME="E1:17_2_1_1_2_1_1_2_14_2_1_1_1"/>
 ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c14">G</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_2_1_1_2_1_1_2_14_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="limfunc3.html#E7:17_2_1_1_2_1_1_2_14"><i><font color="Green" title="E52">A79</font></i></a>, <a class="txt" href="limfunc3.html#E2:17_2_1_1_2_1_1_2_14_2"><i><font color="Green" title="E54">A81</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_14_2_1_1_2"><b>suppose </b></a><a NAME="E1:17_2_1_1_2_1_1_2_14_2_1_1_2"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c14">G</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_2_1_1_2_1_1_2_14_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="limfunc3.html#E3:17_2_1_1_2_1_1_2_14"><i><font color="Green" title="E49">A75</font></i></a>, <a class="txt" href="limfunc3.html#E2:17_2_1_1_2_1_1_2_14_2"><i><font color="Green" title="E54">A81</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:17_2_1_1_2_1_1_2_14_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c17">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E10:17_2_1_1_2_1_1_2_14"/><i><font color="Green" title="E54">A82</font></i>: 
<font color="Maroon" title="c14">G</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">NX</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E7:17_2_1_1_2_1_1_2_14"><i><font color="Green" title="E52">A79</font></i></a></i>;<br/>
<a NAME="E11:17_2_1_1_2_1_1_2_14"/><i><font color="Green" title="E55">A83</font></i>: 
( <font color="Maroon" title="c16">NX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">G</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c16">NX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font> holds <br/><font color="Maroon" title="c14">G</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> ) )
 
<b>by </b><i><a class="txt" href="limfunc3.html#E37:17_2_1_1_2_1_1_2"><i><font color="Green" title="E44">A70</font></i></a>, <a class="txt" href="limfunc3.html#E10:17_2_1_1_2_1_1_2_14"><i><font color="Green" title="E54">A82</font></i></a></i>;<br/>
<a NAME="E12:17_2_1_1_2_1_1_2_14"/><i><font color="Green" title="E56">A84</font></i>: 
<font color="Maroon" title="c14">G</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">N1</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E37:17_2_1_1_2_1_1_2"><i><font color="Green" title="E44">A70</font></i></a>, <a class="txt" href="limfunc3.html#E3:17_2_1_1_2_1_1_2_14"><i><font color="Green" title="E49">A75</font></i></a>, <a class="txt" href="limfunc3.html#E7:17_2_1_1_2_1_1_2_14"><i><font color="Green" title="E52">A79</font></i></a>, <a class="txt" href="limfunc3.html#E10:17_2_1_1_2_1_1_2_14"><i><font color="Green" title="E54">A82</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_14_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_2_1_1_2_1_1_2_14_3"/>
<font color="Maroon" title="c14">G</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">N1</font>
 ;<br/><a NAME="E2:17_2_1_1_2_1_1_2_14_3"/><b>then </b>
<font color="Maroon" title="c14">G</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">N1</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E12:17_2_1_1_2_1_1_2_14"><i><font color="Green" title="E56">A84</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:17_2_1_1_2_1_1_2_14_3"/>
contradiction
 <b>by </b><i><a class="txt" href="limfunc3.html#E8:17_2_1_1_2_1_1_2_14"><i><font color="Green" title="E53">A80</font></i></a>, <a class="txt" href="limfunc3.html#E11:17_2_1_1_2_1_1_2_14"><i><font color="Green" title="E55">A83</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:17_2_1_1_2_1_1_2_14"/>
contradiction
 <b>by </b><i><a class="txt" href="limfunc3.html#E3:17_2_1_1_2_1_1_2_14"><i><font color="Green" title="E49">A75</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E45:17_2_1_1_2_1_1_2"/><i><font color="Green" title="E48">A85</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c14">G</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_15"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>8</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c14">G</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font>;<br/>
<a NAME="E1:17_2_1_1_2_1_1_2_15"/><i><font color="Green" title="E49">A86</font></i>: 
<font color="Maroon">S<sub>8</sub></font>[0]
 
<b>by </b><i><a class="txt" href="limfunc3.html#E32:17_2_1_1_2_1_1_2"><i><font color="Green" title="E41">A64</font></i></a>, <a class="txt" href="limfunc3.html#E37:17_2_1_1_2_1_1_2"><i><font color="Green" title="E44">A70</font></i></a>, <a class="ref" href="seqm_3.html#T31" title="SEQM_3:th.31">SEQM_3:31</a></i>;<br/>
<a NAME="E2:17_2_1_1_2_1_1_2_15"/><i><font color="Green" title="E50">A87</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>8</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>8</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:17_2_1_1_2_1_1_2_15_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c14">G</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font>
 ;<br/>

<a NAME="E2:17_2_1_1_2_1_1_2_15_1"/>
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c14">G</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c14">G</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>]
 
<b>by </b><i><a class="txt" href="limfunc3.html#E37:17_2_1_1_2_1_1_2"><i><font color="Green" title="E44">A70</font></i></a></i>;<br/>
<a NAME="E3:17_2_1_1_2_1_1_2_15_1"/><b>then </b>
<font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font>
 
;<br/>
<b>hence </b><a NAME="E4:17_2_1_1_2_1_1_2_15_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c14">G</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#T31" title="SEQM_3:th.31">SEQM_3:31</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:17_2_1_1_2_1_1_2_15"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>8</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="limfunc3.html#E1:17_2_1_1_2_1_1_2_15"><i><font color="Green" title="E49">A86</font></i></a>, <a class="txt" href="limfunc3.html#E2:17_2_1_1_2_1_1_2_15"><i><font color="Green" title="E50">A87</font></i></a>);</i><br/>


</div><b>end;</b></div><a NAME="E46:17_2_1_1_2_1_1_2"/><i><font color="Green" title="E49">A88</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c14">G</font> is    <a href="seqm_3.html#M1" title="SEQM_3:mode.1">subsequence</a> of <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D10" title="SEQM_3:def.10">SEQM_3:def 10</a></i>;<br/><a NAME="E47:17_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E50">A89</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c14">G</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:17_2"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="seq_4.html#T29" title="SEQ_4:th.29">SEQ_4:29</a></i>;<br/><a NAME="E48:17_2_1_1_2_1_1_2"/><i><font color="Green" title="E51">A90</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c14">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:17_2"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="limfunc3.html#E46:17_2_1_1_2_1_1_2"><i><font color="Green" title="E49">A88</font></i></a>, <a class="ref" href="seq_4.html#T30" title="SEQ_4:th.30">SEQ_4:30</a></i>;<br/><a NAME="E49:17_2_1_1_2_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c14">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E46:17_2_1_1_2_1_1_2"><i><font color="Green" title="E49">A88</font></i></a>, <a class="ref" href="rfunct_2.html#T11" title="RFUNCT_2:th.11">RFUNCT_2:11</a></i>;<br/><a NAME="E50:17_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E52">A91</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c14">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:17_2"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E51:17_2_1_1_2_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c14">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c1">x0</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_16"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_2_1_1_2_1_1_2_16"/><i><font color="Green" title="E53">A92</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c14">G</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:17_2_1_1_2_1_1_2_16"/><i><font color="Green" title="E54">A93</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c14">G</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9" title="RFUNCT_2:th.9">RFUNCT_2:9</a></i>;<br/>
<a NAME="E4:17_2_1_1_2_1_1_2_16"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c14">G</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E45:17_2_1_1_2_1_1_2"><i><font color="Green" title="E48">A85</font></i></a></i>;<br/>
<a NAME="E5:17_2_1_1_2_1_1_2_16"/><b>then </b>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g1</font> where <font color="Olive" title="b1">g1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g1</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="limfunc3.html#E3:17_2_1_1_2_1_1_2_16"><i><font color="Green" title="E54">A93</font></i></a></i>;<br/>
<a NAME="E6:17_2_1_1_2_1_1_2_16"/><b>then </b><i><font color="Green" title="E55">A94</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c1">x0</font>
 
<b>by </b><i><a class="ref" href="limfunc1.html#T152" title="LIMFUNC1:th.152">LIMFUNC1:152</a></i>;<br/>
<a NAME="E7:17_2_1_1_2_1_1_2_16"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E50:17_2_1_1_2_1_1_2"><i><font color="Green" title="E52">A91</font></i></a>, <a class="txt" href="limfunc3.html#E1:17_2_1_1_2_1_1_2_16"><i><font color="Green" title="E53">A92</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:17_2_1_1_2_1_1_2_16"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c1">x0</font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc3.html#E6:17_2_1_1_2_1_1_2_16"><i><font color="Green" title="E55">A94</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E52:17_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E53">A95</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c14">G</font></span>)</span> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 <b>by </b><i><a class="txt" href="limfunc3.html#E2:17"><i><font color="Green" title="E17">A7</font></i></a>, <a class="txt" href="limfunc3.html#E47:17_2_1_1_2_1_1_2"><i><font color="Green" title="E50">A89</font></i></a>, <a class="txt" href="limfunc3.html#E48:17_2_1_1_2_1_1_2"><i><font color="Green" title="E51">A90</font></i></a>, <a class="ref" href="limfunc2.html#D5" title="LIMFUNC2:def.5">LIMFUNC2:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_17"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>consider </b><font color="Maroon" title="c16">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E2:17_2_1_1_2_1_1_2_17"/><i><font color="Green" title="E54">A96</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c16">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon" title="c15">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c9">F</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E27:17_2_1_1_2_1_1_2"><i><font color="Green" title="E38">A60</font></i></a>, <a class="ref" href="limfunc1.html#D4" title="LIMFUNC1:def.4">LIMFUNC1:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">n2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:17_2_1_1_2_1_1_2_17"/><i><font color="Green" title="E55">A97</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c17">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon" title="c15">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c14">G</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E52:17_2_1_1_2_1_1_2"><i><font color="Green" title="E53">A95</font></i></a>, <a class="ref" href="limfunc1.html#D4" title="LIMFUNC1:def.4">LIMFUNC1:def 4</a></i>;<br/><b>take </b><font color="Maroon" title="c18">n</font> =  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Maroon" title="c16">n1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Maroon" title="c17">n2</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c19">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E5:17_2_1_1_2_1_1_2_17"/>
<font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9" title="RFUNCT_2:th.9">RFUNCT_2:9</a></i>;<br/><a NAME="E6:17_2_1_1_2_1_1_2_17"/><b>then </b>
not <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:17_2"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E7:17_2_1_1_2_1_1_2_17"/><b>then </b><i><font color="Green" title="E56">A98</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>assume </b><a NAME="E8:17_2_1_1_2_1_1_2_17"/><i><font color="Green" title="E57">A99</font></i>: 
<font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_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="17_2_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="E1:17_2_1_1_2_1_1_2_17_1_1"/>
( <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> or <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font> )
 </span><b>by </b><i><a class="txt" href="limfunc3.html#E7:17_2_1_1_2_1_1_2_17"><i><font color="Green" title="E56">A98</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_17_1_1_1"><b>suppose </b></a><a NAME="E1:17_2_1_1_2_1_1_2_17_1_1_1"/>
<font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c20">l</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:17_2_1_1_2_1_1_2_17_1_1_1"/><i><font color="Green" title="E58">A100</font></i>: 
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Maroon" title="c20">l</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E19:17_2_1_1_2_1_1_2"><i><font color="Green" title="E32">A38</font></i></a></i>;<br/><a NAME="E4:17_2_1_1_2_1_1_2_17_1_1_1"/>
<font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Maroon" title="c16">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">n</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/><a NAME="E5:17_2_1_1_2_1_1_2_17_1_1_1"/><b>then </b><i><font color="Green" title="E59">A101</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Maroon" title="c16">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E8:17_2_1_1_2_1_1_2_17"><i><font color="Green" title="E57">A99</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:17_2_1_1_2_1_1_2_17_1_1_1"/><i><font color="Green" title="E60">A102</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E25:17_2_1_1_2_1_1_2"><i><font color="Green" title="E37">A56</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E7:17_2_1_1_2_1_1_2_17_1_1_1"/><i><font color="Green" title="E61">A103</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:17_2"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:17_2_1_1_2_1_1_2_17_1_1_1"/>
<font color="Maroon" title="c20">l</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c16">n1</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E3:17_2_1_1_2_1_1_2_17_1_1_1"><i><font color="Green" title="E58">A100</font></i></a>, <a class="txt" href="limfunc3.html#E5:17_2_1_1_2_1_1_2_17_1_1_1"><i><font color="Green" title="E59">A101</font></i></a>, <a class="ref" href="seqm_3.html#T7" title="SEQM_3:th.7">SEQM_3:7</a></i>;<br/><a NAME="E9:17_2_1_1_2_1_1_2_17_1_1_1"/><b>then </b>
<font color="Maroon" title="c15">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c9">F</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c20">l</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E2:17_2_1_1_2_1_1_2_17"><i><font color="Green" title="E54">A96</font></i></a></i>;<br/><a NAME="E10:17_2_1_1_2_1_1_2_17_1_1_1"/><b>then </b>
<font color="Maroon" title="c15">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c20">l</font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc3.html#E6:17_2_1_1_2_1_1_2_17_1_1_1"><i><font color="Green" title="E60">A102</font></i></a>, <a class="ref" href="rfunct_2.html#T21" title="RFUNCT_2:th.21">RFUNCT_2:21</a></i>;<br/><a NAME="E11:17_2_1_1_2_1_1_2_17_1_1_1"/><b>then </b>
<font color="Maroon" title="c15">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc3.html#E3:17_2_1_1_2_1_1_2_17_1_1_1"><i><font color="Green" title="E58">A100</font></i></a>, <a class="ref" href="seqm_3.html#T31" title="SEQM_3:th.31">SEQM_3:31</a></i>;<br/><b>hence </b><a NAME="E12:17_2_1_1_2_1_1_2_17_1_1_1"/>
<font color="Maroon" title="c15">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E7:17_2_1_1_2_1_1_2_17_1_1_1"><i><font color="Green" title="E61">A103</font></i></a>, <a class="ref" href="rfunct_2.html#T21" title="RFUNCT_2:th.21">RFUNCT_2:21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_2_1_1_2_17_1_1_2"><b>suppose </b></a><a NAME="E1:17_2_1_1_2_1_1_2_17_1_1_2"/>
<font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">x0</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c20">l</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:17_2_1_1_2_1_1_2_17_1_1_2"/><i><font color="Green" title="E58">A104</font></i>: 
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">G</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Maroon" title="c20">l</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E44:17_2_1_1_2_1_1_2"><i><font color="Green" title="E47">A73</font></i></a></i>;<br/><a NAME="E4:17_2_1_1_2_1_1_2_17_1_1_2"/>
<font color="Maroon" title="c14">G</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Maroon" title="c17">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">n</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/><a NAME="E5:17_2_1_1_2_1_1_2_17_1_1_2"/><b>then </b><i><font color="Green" title="E59">A105</font></i>: 
<font color="Maroon" title="c14">G</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Maroon" title="c17">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E8:17_2_1_1_2_1_1_2_17"><i><font color="Green" title="E57">A99</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:17_2_1_1_2_1_1_2_17_1_1_2"/><i><font color="Green" title="E60">A106</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c14">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E50:17_2_1_1_2_1_1_2"><i><font color="Green" title="E52">A91</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E7:17_2_1_1_2_1_1_2_17_1_1_2"/><i><font color="Green" title="E61">A107</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:17_2"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:17_2_1_1_2_1_1_2_17_1_1_2"/>
<font color="Maroon" title="c20">l</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c17">n2</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E3:17_2_1_1_2_1_1_2_17_1_1_2"><i><font color="Green" title="E58">A104</font></i></a>, <a class="txt" href="limfunc3.html#E5:17_2_1_1_2_1_1_2_17_1_1_2"><i><font color="Green" title="E59">A105</font></i></a>, <a class="ref" href="seqm_3.html#T7" title="SEQM_3:th.7">SEQM_3:7</a></i>;<br/><a NAME="E9:17_2_1_1_2_1_1_2_17_1_1_2"/><b>then </b>
<font color="Maroon" title="c15">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c14">G</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c20">l</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E4:17_2_1_1_2_1_1_2_17"><i><font color="Green" title="E55">A97</font></i></a></i>;<br/><a NAME="E10:17_2_1_1_2_1_1_2_17_1_1_2"/><b>then </b>
<font color="Maroon" title="c15">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c14">G</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c20">l</font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc3.html#E6:17_2_1_1_2_1_1_2_17_1_1_2"><i><font color="Green" title="E60">A106</font></i></a>, <a class="ref" href="rfunct_2.html#T21" title="RFUNCT_2:th.21">RFUNCT_2:21</a></i>;<br/><a NAME="E11:17_2_1_1_2_1_1_2_17_1_1_2"/><b>then </b>
<font color="Maroon" title="c15">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc3.html#E3:17_2_1_1_2_1_1_2_17_1_1_2"><i><font color="Green" title="E58">A104</font></i></a>, <a class="ref" href="seqm_3.html#T31" title="SEQM_3:th.31">SEQM_3:31</a></i>;<br/><b>hence </b><a NAME="E12:17_2_1_1_2_1_1_2_17_1_1_2"/>
<font color="Maroon" title="c15">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E7:17_2_1_1_2_1_1_2_17_1_1_2"><i><font color="Green" title="E61">A107</font></i></a>, <a class="ref" href="rfunct_2.html#T21" title="RFUNCT_2:th.21">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="E10:17_2_1_1_2_1_1_2_17"/>
<font color="Maroon" title="c15">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">k</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E54:17_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">s</font> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 <b>by </b><i><a class="ref" href="limfunc1.html#D4" title="LIMFUNC1:def.4">LIMFUNC1:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_2_1_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">s</font> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">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="E3:17_2"/>
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c3">s</font> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:17"/>
<font color="Maroon" title="c2">f</font> <a href="limfunc3.html#R2" title="LIMFUNC3:pred.2">is_divergent_to+infty_in</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E5:17"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="limfunc3.html#D2" target="_self" title="LIMFUNC3:def.2">Def2</a></i>;<br/>


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