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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">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:63"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">Y2</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">k2</font></span>)</span> where <font color="Olive">k2</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">k2</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/>
<a NAME="E3:63"/><i><font color="Green">E39</font></i>: 
<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="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y2</font>
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E3:63"><i><font color="Green">E39</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</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">k1</font></span>)</span> where <font color="Olive">k1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k1</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/><a NAME="E2:63_1"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">Y1</font>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E3:63_1"/>
( <font color="Maroon">Y1</font> is <a href="seq_4.html#V3">bounded</a> &amp; <font color="Maroon">Y2</font> is <a href="seq_4.html#V3">bounded</a> )
 <b>by </b><i><a class="ref" href="rinfsup1.html#T11" target="_self">Th11</a>, </i>;<br/><a NAME="E4:63_1"/><b>then </b><i><font color="Green">E41</font></i>: 
( <font color="Maroon">Y1</font> is <a href="seq_4.html#V2">bounded_below</a> &amp; <font color="Maroon">Y2</font> is <a href="seq_4.html#V1">bounded_above</a> )
 <b>by </b><i><a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/><a NAME="E5:63_1"/>
( <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">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:63_1"/><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">m</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font> &amp; <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">m</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font> )
 ;<br/><a NAME="E7:63_1"/><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">m</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">Y1</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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">m</font></span>)</span> )
 <b>by </b><i><a class="ref" href="rinfsup1.html#T14" target="_self">Th14</a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><b>hence </b><a NAME="E8:63_1"/>
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T13" target="_self">Th13</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="E5:63"/>
 <a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <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>
 <b>by </b><i>, <a class="ref" href="rinfsup1.html#T12" target="_self">Th12</a></i>;<br/>


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