<?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="c1">seq1</font> is  <a href="rinfsup1.html#NV1" title="RINFSUP1:NV.1">nonnegative</a>  &amp; <font color="Maroon" title="c2">seq2</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="rinfsup1.html#NV1" title="RINFSUP1:NV.1">nonnegative</a>  implies ( <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> &amp;  <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c2">seq2</font></span>)</span> ) )</span><br/>



<a NAME="E1:102"/><span class="lab"><font color="Green" title="E77">A1</font></span>: 
for <font color="Olive" title="b1">seq1</font>, <font color="Olive" title="b2">seq2</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st <font color="Olive" title="b1">seq1</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Olive" title="b1">seq1</font> is  <a href="rinfsup1.html#NV1" title="RINFSUP1:NV.1">nonnegative</a>  &amp; <font color="Olive" title="b2">seq2</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Olive" title="b2">seq2</font> is  <a href="rinfsup1.html#NV1" title="RINFSUP1:NV.1">nonnegative</a>  holds <br/> <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Olive" title="b2">seq2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Olive" title="b1">seq1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Olive" title="b2">seq2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">seq1</font>, <font color="Maroon" title="c4">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="c3">seq1</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Maroon" title="c3">seq1</font> is  <a href="rinfsup1.html#NV1" title="RINFSUP1:NV.1">nonnegative</a>  &amp; <font color="Maroon" title="c4">seq2</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Maroon" title="c4">seq2</font> is  <a href="rinfsup1.html#NV1" title="RINFSUP1:NV.1">nonnegative</a>  implies  <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c4">seq2</font></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:102_1"/><span class="lab"><font color="Green" title="E78">A2</font></span>: 
( <font color="Maroon" title="c3">seq1</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Maroon" title="c3">seq1</font> is  <a href="rinfsup1.html#NV1" title="RINFSUP1:NV.1">nonnegative</a>  )
 <span class="kw">and </span><br/><a NAME="E2:102_1"/><span class="lab"><font color="Green" title="E79">A3</font></span>: 
( <font color="Maroon" title="c4">seq2</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Maroon" title="c4">seq2</font> is  <a href="rinfsup1.html#NV1" title="RINFSUP1:NV.1">nonnegative</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">seq3</font> =  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c3">seq1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">seq4</font> =  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c4">seq2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">seq5</font> =  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span>;<br/>
<a NAME="E3:102_1"/><span class="lab"><font color="Green" title="E80">A4</font></span>: 
 <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102_1"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="txt" href="#E2:102_1"><span class="lab"><font color="Green" title="E79">A3</font></span></a>, <a class="ref" href="rinfsup1.html#T60" target="_self" title="RINFSUP1:th.60">Th60</a></span>;<br/>
<a NAME="E4:102_1"/><span class="lab"><font color="Green" title="E81">A5</font></span>: 
(  <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">inf</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> &amp;  <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">inf</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102_1"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="txt" href="#E2:102_1"><span class="lab"><font color="Green" title="E79">A3</font></span></a>, <a class="ref" href="rinfsup1.html#T60" target="_self" title="RINFSUP1:th.60">Th60</a></span>;<br/>
<a NAME="E5:102_1"/><span class="lab"><font color="Green" title="E82">A6</font></span>: 
 <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">inf</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:102_1"><span class="lab"><font color="Green" title="E79">A3</font></span></a>, <a class="ref" href="rinfsup1.html#T60" target="_self" title="RINFSUP1:th.60">Th60</a></span>;<br/>
<a NAME="E6:102_1"/><span class="lab"><font color="Green" title="E83">A7</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  <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></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"><span class="p2">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font></span><br/>

<a NAME="E1:102_1_1"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102_1"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="txt" href="#E2:102_1"><span class="lab"><font color="Green" title="E79">A3</font></span></a>, <a class="ref" href="rinfsup1.html#T82" target="_self" title="RINFSUP1:th.82">Th82</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:102_1_1"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</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="E7:102_1"/><span class="lab"><font color="Green" title="E84">A8</font></span>: 
(  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c3">seq1</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c4">seq2</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102_1"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="txt" href="#E2:102_1"><span class="lab"><font color="Green" title="E79">A3</font></span></a>, <a class="ref" href="rinfsup1.html#T60" target="_self" title="RINFSUP1:th.60">Th60</a></span>;<br/>
<a NAME="E8:102_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#T28" title="SEQ_2:th.28">SEQ_2:28</a></span>;<br/>
<a NAME="E9:102_1"/><span class="kw">then </span>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:102_1"><span class="lab"><font color="Green" title="E80">A4</font></span></a>, <a class="txt" href="#E6:102_1"><span class="lab"><font color="Green" title="E83">A7</font></span></a>, <a class="ref" href="seq_2.html#T32" title="SEQ_2:th.32">SEQ_2:32</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:102_1"/>
 <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c4">seq2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:102_1"><span class="lab"><font color="Green" title="E84">A8</font></span></a>, <a class="txt" href="#E4:102_1"><span class="lab"><font color="Green" title="E81">A5</font></span></a>, <a class="txt" href="#E5:102_1"><span class="lab"><font color="Green" title="E82">A6</font></span></a>, <a class="ref" href="seq_2.html#T29" title="SEQ_2:th.29">SEQ_2:29</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="E2:102"/>
for <font color="Olive" title="b1">seq1</font>, <font color="Olive" title="b2">seq2</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st <font color="Olive" title="b1">seq1</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Olive" title="b1">seq1</font> is  <a href="rinfsup1.html#NV1" title="RINFSUP1:NV.1">nonnegative</a>  &amp; <font color="Olive" title="b2">seq2</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Olive" title="b2">seq2</font> is  <a href="rinfsup1.html#NV1" title="RINFSUP1:NV.1">nonnegative</a>  holds <br/><span class="p1">(<span class="default"><a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Olive" title="b1">seq1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Olive" title="b2">seq2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Olive" title="b2">seq2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">seq1</font>, <font color="Maroon" title="c4">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="c3">seq1</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Maroon" title="c3">seq1</font> is  <a href="rinfsup1.html#NV1" title="RINFSUP1:NV.1">nonnegative</a>  &amp; <font color="Maroon" title="c4">seq2</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Maroon" title="c4">seq2</font> is  <a href="rinfsup1.html#NV1" title="RINFSUP1:NV.1">nonnegative</a>  implies <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c4">seq2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:102_2"/><span class="lab"><font color="Green" title="E78">A9</font></span>: 
( <font color="Maroon" title="c3">seq1</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Maroon" title="c3">seq1</font> is  <a href="rinfsup1.html#NV1" title="RINFSUP1:NV.1">nonnegative</a>  )
 <span class="kw">and </span><br/><a NAME="E2:102_2"/><span class="lab"><font color="Green" title="E79">A10</font></span>: 
( <font color="Maroon" title="c4">seq2</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Maroon" title="c4">seq2</font> is  <a href="rinfsup1.html#NV1" title="RINFSUP1:NV.1">nonnegative</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#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c4">seq2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">seq3</font> =  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">seq1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">seq4</font> =  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c4">seq2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">seq5</font> =  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span>;<br/>
<a NAME="E3:102_2"/><span class="lab"><font color="Green" title="E80">A11</font></span>: 
 <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102_2"><span class="lab"><font color="Green" title="E78">A9</font></span></a>, <a class="txt" href="#E2:102_2"><span class="lab"><font color="Green" title="E79">A10</font></span></a>, <a class="ref" href="rinfsup1.html#T59" target="_self" title="RINFSUP1:th.59">Th59</a></span>;<br/>
<a NAME="E4:102_2"/><span class="lab"><font color="Green" title="E81">A12</font></span>: 
(  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> &amp;  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102_2"><span class="lab"><font color="Green" title="E78">A9</font></span></a>, <a class="txt" href="#E2:102_2"><span class="lab"><font color="Green" title="E79">A10</font></span></a>, <a class="ref" href="rinfsup1.html#T59" target="_self" title="RINFSUP1:th.59">Th59</a></span>;<br/>
<a NAME="E5:102_2"/><span class="lab"><font color="Green" title="E82">A13</font></span>: 
 <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:102_2"><span class="lab"><font color="Green" title="E79">A10</font></span></a>, <a class="ref" href="rinfsup1.html#T59" target="_self" title="RINFSUP1:th.59">Th59</a></span>;<br/>
<a NAME="E6:102_2"/><span class="lab"><font color="Green" title="E83">A14</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  <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></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#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">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#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font></span><br/>

<a NAME="E1:102_2_1"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102_2"><span class="lab"><font color="Green" title="E78">A9</font></span></a>, <a class="txt" href="#E2:102_2"><span class="lab"><font color="Green" title="E79">A10</font></span></a>, <a class="ref" href="rinfsup1.html#T81" target="_self" title="RINFSUP1:th.81">Th81</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:102_2_1"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</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="E7:102_2"/><span class="lab"><font color="Green" title="E84">A15</font></span>: 
(  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">seq1</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c4">seq2</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102_2"><span class="lab"><font color="Green" title="E78">A9</font></span></a>, <a class="txt" href="#E2:102_2"><span class="lab"><font color="Green" title="E79">A10</font></span></a>, <a class="ref" href="rinfsup1.html#T59" target="_self" title="RINFSUP1:th.59">Th59</a></span>;<br/>
<a NAME="E8:102_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#T28" title="SEQ_2:th.28">SEQ_2:28</a></span>;<br/>
<a NAME="E9:102_2"/><span class="kw">then </span>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:102_2"><span class="lab"><font color="Green" title="E80">A11</font></span></a>, <a class="txt" href="#E6:102_2"><span class="lab"><font color="Green" title="E83">A14</font></span></a>, <a class="ref" href="seq_2.html#T32" title="SEQ_2:th.32">SEQ_2:32</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:102_2"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c4">seq2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:102_2"><span class="lab"><font color="Green" title="E84">A15</font></span></a>, <a class="txt" href="#E4:102_2"><span class="lab"><font color="Green" title="E81">A12</font></span></a>, <a class="txt" href="#E5:102_2"><span class="lab"><font color="Green" title="E82">A13</font></span></a>, <a class="ref" href="seq_2.html#T29" title="SEQ_2:th.29">SEQ_2:29</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="E3:102"/>
( <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="c1">seq1</font> is  <a href="rinfsup1.html#NV1" title="RINFSUP1:NV.1">nonnegative</a>  &amp; <font color="Maroon" title="c2">seq2</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="rinfsup1.html#NV1" title="RINFSUP1:NV.1">nonnegative</a>  implies ( <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> &amp;  <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c2">seq2</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102"><span class="lab"><font color="Green" title="E77">A1</font></span></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>
