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

<b>let </b><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>;<br/>



<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E18">A1</font></i>: 
( <font color="Maroon" title="c1">seq1</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> &amp; <font color="Maroon" title="c2">seq2</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> )
 ;<br/>

<a NAME="E2:19"/>
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"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c2">seq2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:19_1"/><i><font color="Green" title="E19">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>;<br/>
<a NAME="E2:19_1"/>
( <font color="Maroon" title="c1">seq1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c1">seq1</font> &amp; <font color="Maroon" title="c2">seq2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c2">seq2</font> )
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:19"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="rinfsup1.html#T7" target="_self" title="RINFSUP1:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E3:19_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c2">seq2</font></span>)</span>
 <b>by </b><i><a class="txt" href="rinfsup1.html#E1:19_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:19"/>
 <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c2">seq2</font></span>)</span>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T9" target="_self" title="RINFSUP1:th.9">Th9</a></i>;<br/>


</div>
