<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">seq1</font>, <font color="Maroon" title="c2">seq2</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">seq1</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Maroon" title="c2">seq2</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c1">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">seq2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> ) implies ( ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> ) &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> ) ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:83"/><span class="lab"><font color="Green" title="E69">A1</font></span>: 
<font color="Maroon" title="c1">seq1</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 <span class="kw">and </span><br/><a NAME="E2:83"/><span class="lab"><font color="Green" title="E70">A2</font></span>: 
<font color="Maroon" title="c2">seq2</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 <span class="kw">and </span><br/><a NAME="E3:83"/><span class="lab"><font color="Green" title="E71">A3</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c1">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">seq2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> ) &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> ) )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> &amp; <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c4">Y2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k2</font></span>)</span> where <font color="Olive" title="b1">k2</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k2</font> </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup1.html#T29" target="_self" title="RINFSUP1:th.29">Th29</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">Y1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k1</font></span>)</span> where <font color="Olive" title="b1">k1</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k1</font> </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup1.html#T29" target="_self" title="RINFSUP1:th.29">Th29</a></span>;<br/><a NAME="E3:83_1"/><span class="lab"><font color="Green" title="E72">A4</font></span>: 
( <font color="Maroon" title="c5">Y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">Y2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_1.html#T1" title="SETLIM_1:th.1">SETLIM_1:1</a></span>;<br/><a NAME="E4:83_1"/><span class="lab"><font color="Green" title="E73">A5</font></span>: 
( <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c5">Y1</font> &amp; <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c4">Y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup1.html#D4" target="_self" title="RINFSUP1:def.4">Def4</a></span>;<br/><a NAME="E5:83_1"/>
<font color="Maroon" title="c5">Y1</font> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="ref" href="rinfsup1.html#T33" target="_self" title="RINFSUP1:th.33">Th33</a></span>;<br/><a NAME="E6:83_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A6</font></span>: 
<font color="Maroon" title="c5">Y1</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D11" title="XXREAL_2:def.11">XXREAL_2:def 11</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">r</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y2</font> implies  <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c5">Y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font> )</span><br/><span class="kw">assume </span><a NAME="E1:83_1_1"/>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c5">Y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:83_1_1"/><span class="lab"><font color="Green" title="E75">A7</font></span>: 
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 <span class="kw">and </span><br/><a NAME="E4:83_1_1"/><span class="lab"><font color="Green" title="E76">A8</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font>
 ;<br/><a NAME="E5:83_1_1"/>
<font color="Maroon" title="c1">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_1_1"><span class="lab"><font color="Green" title="E76">A8</font></span></a></span>;<br/><a NAME="E6:83_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A9</font></span>: 
 <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c5">Y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_1"><span class="lab"><font color="Green" title="E74">A6</font></span></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></span>;<br/><a NAME="E7:83_1_1"/>
<font color="Maroon" title="c1">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="txt" href="#E3:83_1_1"><span class="lab"><font color="Green" title="E75">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:83_1_1"/>
 <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c5">Y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_1_1"><span class="lab"><font color="Green" title="E77">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E8:83_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A10</font></span>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y2</font> holds <br/> <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c5">Y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font>
 ;<br/><a NAME="E9:83_1"/>
<font color="Maroon" title="c4">Y2</font> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="rinfsup1.html#T33" target="_self" title="RINFSUP1:th.33">Th33</a></span>;<br/><a NAME="E10:83_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A11</font></span>: 
<font color="Maroon" title="c4">Y2</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D11" title="XXREAL_2:def.11">XXREAL_2:def 11</a></span>;<br/><div><span class="lab"><font color="Green" title="E77">A12</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">r</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y1</font> implies <font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c4">Y2</font> )</span><br/><span class="kw">assume </span><a NAME="E1:83_1_2"/>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c4">Y2</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:83_1_2"/><span class="lab"><font color="Green" title="E78">A13</font></span>: 
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 <span class="kw">and </span><br/><a NAME="E4:83_1_2"/><span class="lab"><font color="Green" title="E79">A14</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font>
 ;<br/><a NAME="E5:83_1_2"/>
<font color="Maroon" title="c2">seq2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_1_2"><span class="lab"><font color="Green" title="E79">A14</font></span></a></span>;<br/><a NAME="E6:83_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A15</font></span>: 
<font color="Maroon" title="c2">seq2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c4">Y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:83_1"><span class="lab"><font color="Green" title="E76">A11</font></span></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></span>;<br/><a NAME="E7:83_1_2"/>
<font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">seq2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="txt" href="#E3:83_1_2"><span class="lab"><font color="Green" title="E78">A13</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:83_1_2"/>
<font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c4">Y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_1_2"><span class="lab"><font color="Green" title="E80">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E12:83_1"/>
( <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c5">Y1</font> &amp; <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c4">Y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup1.html#D5" target="_self" title="RINFSUP1:def.5">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E13:83_1"/>
( <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> &amp; <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_1"><span class="lab"><font color="Green" title="E73">A5</font></span></a>, <a class="txt" href="#E3:83_1"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="txt" href="#E11:83_1"><span class="lab"><font color="Green" title="E77">A12</font></span></a>, <a class="txt" href="#E8:83_1"><span class="lab"><font color="Green" title="E75">A10</font></span></a>, <a class="ref" href="seq_4.html#T130" title="SEQ_4:th.130">SEQ_4:130</a>, <a class="ref" href="topmetr3.html#T1" title="TOPMETR3:th.1">TOPMETR3:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:83"/>
( ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> ) &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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