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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>





<b>assume </b><a NAME="E1:91"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V1">bounded_above</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="partfun3.html#V4">nonnegative</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V1">bounded_above</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="partfun3.html#V4">nonnegative</a> )
 ;<br/>

<a NAME="E2:91"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V1">bounded_above</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>1</sub></font> is <a href="partfun3.html#V4">nonnegative</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V1">bounded_above</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>1</sub></font> is <a href="partfun3.html#V4">nonnegative</a> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T26" target="_self">Th26</a>, <a class="ref" href="rinfsup1.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E3:91"/><b>then </b>
 <a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun3.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E4:91"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>1</sub></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="E5:91"/>
( <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T39" target="_self">Th39</a></i>;<br/>
<b>hence </b><a NAME="E6:91"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></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#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rinfsup1.html#E4:91"><i><font color="Green">E77</font></i></a>, <a class="ref" href="rinfsup1.html#T39" target="_self">Th39</a></i>;<br/>


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