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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="partfun3.html#K6">-</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>thus </b><a NAME="E1:14"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V1">bounded_above</a> implies  <a href="partfun3.html#K6">-</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V2">bounded_below</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V1">bounded_above</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:14_1"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="seq_2.html#D3">SEQ_2:def 3</a></i>;<br/>
<a NAME="E4:14_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><a href="partfun3.html#K6">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:14_1_1"/><i><font color="Green">E14</font></i>: 
<span class="p1">(<span class="default"><a href="partfun3.html#K6">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T14">SEQ_1:14</a></i>;<br/>
<a NAME="E2:14_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E3:14_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:14_1_1"/>
<span class="p1">(<span class="default"><a href="partfun3.html#K6">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="rinfsup1.html#E1:14_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:14_1"/>
 <a href="partfun3.html#K6">-</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V2">bounded_below</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D4">SEQ_2:def 4</a></i>;<br/>


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

<b>assume </b><a NAME="E2:14"/>
 <a href="partfun3.html#K6">-</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V2">bounded_below</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:14"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><a href="partfun3.html#K6">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="seq_2.html#D4">SEQ_2:def 4</a></i>;<br/>
<a NAME="E5:14"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:14_2"/>
<span class="p1">(<span class="default"><a href="partfun3.html#K6">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T14">SEQ_1:14</a></i>;<br/>
<a NAME="E2:14_2"/><b>then </b><i><font color="Green">E14</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun3.html#K6">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<a NAME="E3:14_2"/>
<span class="p1">(<span class="default"><a href="partfun3.html#K6">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E4:14"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:14_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="rinfsup1.html#E2:14_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:14"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V1">bounded_above</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D3">SEQ_2:def 3</a></i>;<br/>


</div>
