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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>thus </b><a NAME="E1:13"/>
( ( 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="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> ) implies ( <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V2">bounded_below</a> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1"/><i><font color="Green">E12</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="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:13_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="rinfsup1.html#E1:13_1"><i><font color="Green">E12</font></i></a></i>;<br/><b>hence </b><a NAME="E2:13_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="rinfsup1.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:13_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><a NAME="E2:13_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="rinfsup1.html#E1:13_1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:13_1_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>2</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> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rinfsup1.html#E3:13_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="rinfsup1.html#T8" target="_self">Th8</a></i>;<br/><b>thus </b><a NAME="E5:13_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rinfsup1.html#E1:13_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="rinfsup1.html#E4:13_1_2"><i><font color="Green">E15</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:13_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


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

<b>assume </b><a NAME="E2:13"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V2">bounded_below</a> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:13_2"/>
 <a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="rinfsup1.html#E2:13"><i><font color="Green">E12</font></i></a>, <a class="ref" href="rinfsup1.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><a NAME="E2:13_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="rinfsup1.html#E2:13"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:13"/>
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="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>


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