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<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/><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="E1:89"/>
( <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="E2:89"/><b>then </b>
( <font color="Maroon">seq1</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font> is <a href="seq_2.html#V2">bounded_below</a> &amp; <font color="Maroon">seq1</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font> is <a href="partfun3.html#V4">nonnegative</a> &amp; <font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font> is <a href="seq_2.html#V2">bounded_below</a> &amp; <font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font> is <a href="partfun3.html#V4">nonnegative</a> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:89"/><b>then </b>
 <a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="partfun3.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</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#K2" target="_self">inf</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:89"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</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> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T42">SEQM_3:42</a></i>;<br/>
<a NAME="E5:89"/>
( <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</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>  <a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:89"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</font></span>)</span></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"><span class="p2">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq1</font></span>)</span> <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"><span class="p2">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T27" target="_self">Th27</a>, </i>;<br/>


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