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<b>let </b><font color="Maroon">seq1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">seq2</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:86"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">seq1</font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Maroon">seq2</font> is <a href="seq_2.html#V3">bounded</a> )
 <b>and </b><br/><a NAME="E2:86"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><b>now </b></a><div class="add"><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>reconsider </b><font color="Maroon">Y1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">seq1</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">n</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/><b>reconsider </b><font color="Maroon">Y2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">seq2</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="E3:86_1"/><i><font color="Green">E40</font></i>: 
( <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq1</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">Y1</font> &amp; <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq2</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#E2:86"><i><font color="Green">E39</font></i></a></i>;<br/><a NAME="E4:86_1"/><i><font color="Green">E41</font></i>: 
( <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq1</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#K3">inf</a> <font color="Maroon">Y1</font> &amp; <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq2</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#K3">inf</a> <font color="Maroon">Y2</font> )
 <b>by </b><i><a class="ref" href="rinfsup1.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E5:86_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">Y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Y2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="setlim_1.html#T3">SETLIM_1:3</a></i>;<br/><a NAME="E6:86_1"/><i><font color="Green">E50</font></i>: 
( <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#T27" target="_self">Th27</a>, </i>;<br/><a NAME="E7:86_1"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">Y1</font> is <a href="seq_4.html#V1">bounded_above</a> &amp; <font color="Maroon">Y1</font> is <a href="seq_4.html#V2">bounded_below</a> )
 <b>by </b><i><a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/><a NAME="E8:86_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">Y2</font> is <a href="seq_4.html#V1">bounded_above</a> &amp; <font color="Maroon">Y2</font> is <a href="seq_4.html#V2">bounded_below</a> )
 <b>by </b><i>, <a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/><div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_1"><b>now </b></a><div class="add"><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>assume </b><a NAME="E1:86_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 ;<br/><b>then consider </b><font color="Maroon">k</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="E3:86_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 ;<br/><a NAME="E4:86_1_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T29" target="_self">Th29</a>, <a class="ref" href="rinfsup1.html#T33" target="_self">Th33</a></i>;<br/><a NAME="E5:86_1_1"/>
<font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T33" target="_self">Th33</a></i>;<br/><a NAME="E6:86_1_1"/><b>then </b>
<font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y2</font>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T31" target="_self">Th31</a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E7:86_1_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y2</font>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T34" target="_self">Th34</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_2"><b>now </b></a><div class="add"><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>assume </b><a NAME="E1:86_1_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font>
 ;<br/><b>then consider </b><font color="Maroon">k</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="E3:86_1_2"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 ;<br/><a NAME="E4:86_1_2"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T29" target="_self">Th29</a>, <a class="ref" href="rinfsup1.html#T33" target="_self">Th33</a></i>;<br/><a NAME="E5:86_1_2"/>
<font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T33" target="_self">Th33</a></i>;<br/><a NAME="E6:86_1_2"/><b>then </b>
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">Y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><b>hence </b><a NAME="E7:86_1_2"/>
 <a href="pscomp_1.html#K3">inf</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="ref" href="rinfsup1.html#T34" target="_self">Th34</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><a NAME="E11:86_1"/><b>then </b>
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font> holds <br/> <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">Y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font>
 ;<br/><b>hence </b><a NAME="E12:86_1"/>
( <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <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">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="complsp1.html#T84">COMPLSP1:84</a>, , <a class="ref" href="rinfsup1.html#T32" target="_self">Th32</a>, , <a class="ref" href="rinfsup1.html#T30" target="_self">Th30</a>, <a class="ref" href="topmetr3.html#T1">TOPMETR3:1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:86"/>
( ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <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">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> ) &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> ) )
 ;<br/>


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