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

<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/>



<a NAME="E1:104"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">seq1</font>, <font color="Olive">seq2</font> being  <a href="seq_1.html#NM1">Real_Sequence</a>  st <font color="Olive">seq1</font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Olive">seq2</font> is <a href="seq_2.html#V3">bounded</a> holds <br/><span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Olive">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Olive">seq2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p1">(<span class="default"><font color="Olive">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Olive">seq2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1"><b>proof </b></a><div class="add">

<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>set </b><font color="Maroon">r1</font> =  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font>;<br/>
<b>set </b><font color="Maroon">r2</font> =  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq2</font>;<br/>
<b>assume </b><a NAME="E1:104_1"/><i><font color="Green">E39</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> )
 ;<br/>

<a NAME="E2:104_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T71">SEQM_3:71</a></i>;<br/>
<a NAME="E3:104_1"/>
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">k</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"><span class="p2">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</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="E1:104_1_1"/><i><font color="Green">E41</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E2:104_1_1"/><i><font color="Green">E49</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T35" target="_self">Th35</a>, <a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/>
<b>consider </b><font color="Maroon">n1</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="E4:104_1_1"/><i><font color="Green">E50</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 <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T33" target="_self">Th33</a>, <a class="ref" href="rinfsup1.html#T36" target="_self">Th36</a>, </i>;<br/>
<b>consider </b><font color="Maroon">n2</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="E6:104_1_1"/><i><font color="Green">E51</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 <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T33" target="_self">Th33</a>, <a class="ref" href="rinfsup1.html#T36" target="_self">Th36</a>, </i>;<br/>
<a NAME="E7:104_1_1"/>
for <font color="Olive">k</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"><span class="p2">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:104_1_1_1"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:104_1_1_1"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E3:104_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#D4" target="_self">Def4</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E4:104_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:104_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">k</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"><span class="p2">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:104_1"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T34" target="_self">Th34</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:104"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">seq1</font>, <font color="Olive">seq2</font> being  <a href="seq_1.html#NM1">Real_Sequence</a>  st <font color="Olive">seq1</font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Olive">seq2</font> is <a href="seq_2.html#V3">bounded</a> holds <br/> <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p1">(<span class="default"><font color="Olive">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Olive">seq2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Olive">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Olive">seq2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_2"><b>proof </b></a><div class="add">

<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><a NAME="E1:104_2"/><i><font color="Green">E54</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> )
 ;<br/>

<a NAME="E2:104_2"/><b>then </b><i><font color="Green">E98</font></i>: 
( <font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font> is <a href="seq_2.html#V3">bounded</a> &amp;  <a href="partfun3.html#K6">-</a> <font color="Maroon">seq1</font> is <a href="seq_2.html#V3">bounded</a> &amp;  <a href="partfun3.html#K6">-</a> <font color="Maroon">seq2</font> is <a href="seq_2.html#V3">bounded</a> )
 
<b>by </b><i><a class="ref" href="seqm_3.html#T70">SEQM_3:70</a>, <a class="ref" href="seqm_3.html#T71">SEQM_3:71</a></i>;<br/>
<a NAME="E3:104_2"/>
 <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="partfun3.html#K3">+</a> <span class="p2">(<span class="default"><a href="partfun3.html#K6">-</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p2">(<span class="default"><a href="partfun3.html#K6">-</a> <font color="Maroon">seq2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T35" target="_self">Th35</a>, <a class="ref" href="rinfsup1.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E4:104_2"/><b>then </b>
 <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="partfun3.html#K3">+</a> <span class="p2">(<span class="default"><a href="partfun3.html#K6">-</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T34" target="_self">Th34</a>, <a class="ref" href="rinfsup1.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E5:104_2"/><b>then </b>
 <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="partfun3.html#K4">-</a> <font color="Maroon">seq2</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T15">SEQ_1:15</a></i>;<br/>
<a NAME="E6:104_2"/><b>then </b>
 <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:104_2"/>
 <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:104"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">seq1</font>, <font color="Olive">seq2</font> being  <a href="seq_1.html#NM1">Real_Sequence</a>  st <font color="Olive">seq1</font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Olive">seq2</font> is <a href="seq_2.html#V3">bounded</a> holds <br/> <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p1">(<span class="default"><font color="Olive">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Olive">seq2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Olive">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Olive">seq2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_3"><b>proof </b></a><div class="add">

<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>set </b><font color="Maroon">r1</font> =  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq1</font>;<br/>
<b>set </b><font color="Maroon">r2</font> =  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font>;<br/>
<b>assume </b><a NAME="E1:104_3"/><i><font color="Green">E100</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> )
 ;<br/>

<a NAME="E2:104_3"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T71">SEQM_3:71</a></i>;<br/>
<a NAME="E3:104_3"/>
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">k</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"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</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="E1:104_3_1"/><i><font color="Green">E102</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E2:104_3_1"/><i><font color="Green">E103</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i>, <a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/>
<b>consider </b><font color="Maroon">n1</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="E4:104_3_1"/><i><font color="Green">E104</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">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T35" target="_self">Th35</a>, <a class="ref" href="rinfsup1.html#T37" target="_self">Th37</a>, </i>;<br/>
<b>consider </b><font color="Maroon">n2</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="E6:104_3_1"/><i><font color="Green">E146</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">seq2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T35" target="_self">Th35</a>, <a class="ref" href="rinfsup1.html#T37" target="_self">Th37</a>, </i>;<br/>
<a NAME="E7:104_3_1"/>
for <font color="Olive">k</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"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:104_3_1_1"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E2:104_3_1_1"/>
<font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E3:104_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T38" target="_self">Th38</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E4:104_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:104_3_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">k</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"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:104_3"/>
 <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font></span>)</span>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T36" target="_self">Th36</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:104"/><i><font color="Green">E148</font></i>: 
for <font color="Olive">seq1</font>, <font color="Olive">seq2</font> being  <a href="seq_1.html#NM1">Real_Sequence</a>  st <font color="Olive">seq1</font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Olive">seq2</font> is <a href="seq_2.html#V3">bounded</a> holds <br/><span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Olive">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Olive">seq2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p1">(<span class="default"><font color="Olive">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Olive">seq2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_4"><b>proof </b></a><div class="add">

<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><a NAME="E1:104_4"/><i><font color="Green">E149</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> )
 ;<br/>

<a NAME="E2:104_4"/><b>then </b><i><font color="Green">E150</font></i>: 
( <font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font> is <a href="seq_2.html#V3">bounded</a> &amp;  <a href="partfun3.html#K6">-</a> <font color="Maroon">seq1</font> is <a href="seq_2.html#V3">bounded</a> &amp;  <a href="partfun3.html#K6">-</a> <font color="Maroon">seq2</font> is <a href="seq_2.html#V3">bounded</a> )
 
<b>by </b><i><a class="ref" href="seqm_3.html#T70">SEQM_3:70</a>, <a class="ref" href="seqm_3.html#T71">SEQM_3:71</a></i>;<br/>
<a NAME="E3:104_4"/>
 <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="partfun3.html#K3">+</a> <span class="p2">(<span class="default"><a href="partfun3.html#K6">-</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p2">(<span class="default"><a href="partfun3.html#K6">-</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rinfsup1.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E4:104_4"/><b>then </b>
 <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="partfun3.html#K3">+</a> <span class="p2">(<span class="default"><a href="partfun3.html#K6">-</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T36" target="_self">Th36</a>, <a class="ref" href="rinfsup1.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E5:104_4"/><b>then </b>
 <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq1</font></span>)</span> <a href="partfun3.html#K4">-</a> <font color="Maroon">seq1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T15">SEQ_1:15</a></i>;<br/>
<a NAME="E6:104_4"/><b>then </b>
 <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:104_4"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>


</div><b>end;</b></div>
<b>assume </b><a NAME="E5:104"/><i><font color="Green">E151</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> )
 ;<br/>

<a NAME="E6:104"/><b>then </b><i><font color="Green">E152</font></i>: 
<span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E7:104"/><i><font color="Green">E153</font></i>: 
(  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font></span>)</span> &amp;  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T36" target="_self">Th36</a>, <a class="ref" href="rinfsup1.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E8:104"/><i><font color="Green">E154</font></i>: 
( <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T36" target="_self">Th36</a>, <a class="ref" href="rinfsup1.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E9:104"/><i><font color="Green">E155</font></i>: 
 <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T36" target="_self">Th36</a>, <a class="ref" href="rinfsup1.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E10:104"/>
( ( <font color="Maroon">seq1</font> is <a href="seq_2.html#V4">convergent</a> or <font color="Maroon">seq2</font> is <a href="seq_2.html#V4">convergent</a> ) implies (  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq2</font></span>)</span> &amp;  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:104_5"/><i><font color="Green">E156</font></i>: 
( <font color="Maroon">seq1</font> is <a href="seq_2.html#V4">convergent</a> or <font color="Maroon">seq2</font> is <a href="seq_2.html#V4">convergent</a> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:104_5_1"/>
( <font color="Maroon">seq1</font> is <a href="seq_2.html#V4">convergent</a> or <font color="Maroon">seq2</font> is <a href="seq_2.html#V4">convergent</a> )
 </span><b>by </b><i><a class="ref" href="rinfsup1.html#T38" target="_self">Th38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_5_1_1"><b>suppose </b></a><a NAME="E1:104_5_1_1"/>
<font color="Maroon">seq1</font> is <a href="seq_2.html#V4">convergent</a>
 ;<br/><div class="add"><a NAME="E2:104_5_1_1"/><b>then </b>
 <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font> <a href="hidden.html#R1">=</a>  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:104_5_1_1"/>
(  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq2</font></span>)</span> &amp;  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="rinfsup1.html#T37" target="_self">Th37</a>, <a class="ref" href="rinfsup1.html#D4" target="_self">Def4</a>, <a class="ref" href="rinfsup1.html#D5" target="_self">Def5</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_5_1_2"><b>suppose </b></a><a NAME="E1:104_5_1_2"/>
<font color="Maroon">seq2</font> is <a href="seq_2.html#V4">convergent</a>
 ;<br/><div class="add"><a NAME="E2:104_5_1_2"/><b>then </b>
 <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq2</font> <a href="hidden.html#R1">=</a>  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:104_5_1_2"/>
(  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq2</font></span>)</span> &amp;  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="rinfsup1.html#T37" target="_self">Th37</a>, <a class="ref" href="rinfsup1.html#D4" target="_self">Def4</a>, <a class="ref" href="rinfsup1.html#D5" target="_self">Def5</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E11:104"/>
( <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> &amp;  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font></span>)</span> &amp;  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> &amp;  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font></span>)</span> &amp; ( ( <font color="Maroon">seq1</font> is <a href="seq_2.html#V4">convergent</a> or <font color="Maroon">seq2</font> is <a href="seq_2.html#V4">convergent</a> ) implies (  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq2</font></span>)</span> &amp;  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K3">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq2</font></span>)</span> ) ) )
 <b>by </b><i><a class="ref" href="rinfsup1.html#T36" target="_self">Th36</a>, <a class="ref" href="rinfsup1.html#T32" target="_self">Th32</a>, <a class="ref" href="rinfsup1.html#T33" target="_self">Th33</a>, <a class="ref" href="rinfsup1.html#T34" target="_self">Th34</a>, <a class="ref" href="rinfsup1.html#T35" target="_self">Th35</a></i>;<br/>


</div>
