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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">r</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">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>





<b>assume </b><a NAME="E1:53"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V1">bounded_above</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">Y1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">seq1</font> = <font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font>;<br/>
<a NAME="E3:53"/><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:53"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font> is <a href="seq_2.html#V1">bounded_above</a>
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E5:53"/>
( ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> ) implies <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:53_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/><b>then consider </b><font color="Maroon">r1</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="E3:53_1_1"/><i><font color="Green">E49</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font> )
 <b>by </b><i><a class="txt" href="rinfsup1.html#E5:53_1_1"><i><font color="Green">E50</font></i></a></i>;<br/><b>consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:53_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">k2</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:53_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k2</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E8:53_1_1"/>
<font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E9:53_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="rinfsup1.html#T8" target="_self">Th8</a>, <a class="txt" href="rinfsup1.html#E9:53"><i><font color="Green">E52</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:53_1"/><b>then </b>
 <a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:53_1"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
;<br/>
<a NAME="E5:53_1"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:53_1"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="rinfsup1.html#E3:53"><i><font color="Green">E39</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E6:53"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E7:53"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E3:53"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E8:53"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:53"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:53_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E2:53_2"/>
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 <b>by </b><i><a class="txt" href="rinfsup1.html#E1:53_2"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E3:53_2"/><b>then </b>
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:53_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="rinfsup1.html#E5:53_2"><i><font color="Green">E54</font></i></a></i>;<br/><b>thus </b><a NAME="E7:53_2"/>
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="txt" href="rinfsup1.html#E9:53"><i><font color="Green">E52</font></i></a>, , <a class="txt" href="rinfsup1.html#E5:53_2"><i><font color="Green">E54</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:53"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 ;<br/>


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