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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</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>assume </b><a NAME="E1:77"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="seqm_3.html#V3">non-decreasing</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V1">bounded_above</a> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rinfsup1.html#K1" target="_self">sup</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E2:77"/>
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="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rinfsup1.html#K1" target="_self">sup</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>proof </b></a><div class="add">

<a NAME="E1:77_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:77"><i><font color="Green">E70</font></i></a>, <a class="ref" href="rinfsup1.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E2:77_1"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:77"><i><font color="Green">E70</font></i></a>, <a class="ref" href="rinfsup1.html#T68" target="_self">Th68</a></i>;<br/>
<b>thus </b><a NAME="E3:77_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="rinfsup1.html#E1:77_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="rinfsup1.html#E2:77_1"><i><font color="Green">E72</font></i></a>);</i><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:77"/>
( <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rinfsup1.html#K1" target="_self">sup</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>2</sub></font> is <a href="funct_1.html#V3">constant</a> )
 <b>by </b><i><a class="ref" href="seqm_3.html#D6">SEQM_3:def 6</a></i>;<br/>


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