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<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="E1:25"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">seq1</font> is <a href="seq_2.html#V2">bounded_below</a> &amp; <font color="Maroon">seq1</font> is <a href="partfun3.html#V4">nonnegative</a> &amp; <font color="Maroon">seq2</font> is <a href="seq_2.html#V2">bounded_below</a> &amp; <font color="Maroon">seq2</font> is <a href="partfun3.html#V4">nonnegative</a> )
 ;<br/>

<a NAME="E2:25"/>
for <font color="Olive">n</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#K5">(#)</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</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">seq1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">seq2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:25_1"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>;<br/>
<a NAME="E2:25_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">seq1</font> &amp; <font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">seq2</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:25_1"/>
(  <a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">seq1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">seq2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:25_1"/>
<span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</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">seq1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">seq2</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T68">XREAL_1:68</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:25"/>
( <font color="Maroon">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</font> is <a href="seq_2.html#V2">bounded_below</a> &amp;  <a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K5">(#)</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#K2" target="_self">inf</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">seq2</font></span>)</span> )
 <b>by </b><i/>;<br/>


</div>
