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

<b>let </b><font color="Maroon" title="c1">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>let </b><font color="Maroon" title="c2">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c3">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>





<b>assume </b><a NAME="E1:51"/><i><font color="Green" title="E47">A1</font></i>: 
<font color="Maroon" title="c3">seq</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">Y1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</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>  : <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="rinfsup1.html#T29" target="_self" title="RINFSUP1:th.29">Th29</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">seq1</font> = <font color="Maroon" title="c3">seq</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c1">n</font>;<br/>
<a NAME="E3:51"/><i><font color="Green" title="E48">A2</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Y1</font>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T30" target="_self" title="RINFSUP1:th.30">Th30</a></i>;<br/>
<a NAME="E4:51"/><i><font color="Green" title="E49">A3</font></i>: 
<font color="Maroon" title="c3">seq</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c1">n</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a>
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:51"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="rinfsup1.html#T27" target="_self" title="RINFSUP1:th.27">Th27</a></i>;<br/>
<b>thus </b><a NAME="E5:51"/>
( ( 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" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b1">k</font></span>)</span> ) implies <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">n</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_1"/><i><font color="Green" title="E50">A4</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>  holds  <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b1">k</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">n1</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="E1:51_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">r1</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:51_1_1"/><i><font color="Green" title="E51">A5</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r1</font> &amp; <font color="Maroon" title="c7">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y1</font> )
 <b>by </b><i><a class="txt" href="rinfsup1.html#E3:51"><i><font color="Green" title="E48">A2</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c8">k1</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="E5:51_1_1"/><i><font color="Green" title="E52">A6</font></i>: 
( <font color="Maroon" title="c7">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">k1</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k1</font> )
 <b>by </b><i><a class="txt" href="rinfsup1.html#E3:51_1_1"><i><font color="Green" title="E51">A5</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c9">k2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:51_1_1"/><i><font color="Green" title="E53">A7</font></i>: 
<font color="Maroon" title="c8">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">k2</font>
 <b>by </b><i><a class="txt" href="rinfsup1.html#E5:51_1_1"><i><font color="Green" title="E52">A6</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><a NAME="E8:51_1_1"/>
<font color="Maroon" title="c9">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E9:51_1_1"/>
<font color="Maroon" title="c2">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="c3">seq</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n1</font>
 <b>by </b><i><a class="txt" href="rinfsup1.html#E3:51_1_1"><i><font color="Green" title="E51">A5</font></i></a>, <a class="txt" href="rinfsup1.html#E5:51_1_1"><i><font color="Green" title="E52">A6</font></i></a>, <a class="txt" href="rinfsup1.html#E7:51_1_1"><i><font color="Green" title="E53">A7</font></i></a>, <a class="txt" href="rinfsup1.html#E1:51_1"><i><font color="Green" title="E50">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:51_1"/><b>then </b>
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T10" target="_self" title="RINFSUP1:th.10">Th10</a></i>;<br/>
<a NAME="E4:51_1"/><b>then </b>
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c4">Y1</font>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T30" target="_self" title="RINFSUP1:th.30">Th30</a></i>;<br/>
<b>hence </b><a NAME="E5:51_1"/>
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="rinfsup1.html#D4" target="_self" title="RINFSUP1:def.4">Def4</a></i>;<br/>


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

<b>assume </b><a NAME="E6:51"/>
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<a NAME="E7:51"/><b>then </b>
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c4">Y1</font>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#D4" target="_self" title="RINFSUP1:def.4">Def4</a></i>;<br/>
<a NAME="E8:51"/><b>then </b><i><font color="Green" title="E50">A8</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T30" target="_self" title="RINFSUP1:th.30">Th30</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">m1</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="E1:51_2"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c6">m1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E2:51_2"/><b>then </b>
<font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c6">m1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y1</font>
 ;<br/><a NAME="E3:51_2"/><b>then </b>
<font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c6">m1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T30" target="_self" title="RINFSUP1:th.30">Th30</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:51_2"/><i><font color="Green" title="E51">A9</font></i>: 
( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c6">m1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">k</font> = <font color="Maroon" title="c7">k</font> as    <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="rinfsup1.html#E5:51_2"><i><font color="Green" title="E51">A9</font></i></a></i>;<br/><b>thus </b><a NAME="E7:51_2"/>
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c6">m1</font></span>)</span>
 <b>by </b><i><a class="txt" href="rinfsup1.html#E5:51_2"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="rinfsup1.html#E8:51"><i><font color="Green" title="E50">A8</font></i></a>, <a class="txt" href="rinfsup1.html#E4:51"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="rinfsup1.html#T10" target="_self" title="RINFSUP1:th.10">Th10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:51"/>
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" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b1">k</font></span>)</span>
 ;<br/>


</div>
