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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V2">bounded_below</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V2">bounded_below</a> )
 ;<br/>

<a NAME="E2:18"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="partfun3.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:18_1"/><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="partfun3.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>
<a NAME="E2:18_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:18"><i><font color="Green">E17</font></i></a>, <a class="ref" href="rinfsup1.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E3:18_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="partfun3.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rinfsup1.html#E1:18_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:18"/>
 <a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="partfun3.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T10" target="_self">Th10</a></i>;<br/>


</div>
