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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> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:69"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V1">bounded_above</a>
 ;<br/>

<a NAME="E2:69"/><b>then </b><i><font color="Green">E64</font></i>: 
(  <a href="partfun3.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V2">bounded_below</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="seq_2.html#V1">bounded_above</a> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T26" target="_self">Th26</a>, <a class="ref" href="rinfsup1.html#T11" target="_self">Th11</a></i>;<br/>
<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="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>
<b>by </b><i><a class="ref" href="rinfsup1.html#T39" target="_self">Th39</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p2">(<span class="default"><a href="partfun3.html#K6">-</a> <span class="p3">(<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></span>)</span>
<b>by </b><i><a class="txt" href="rinfsup1.html#E2:69"><i><font color="Green">E64</font></i></a>, <a class="ref" href="rinfsup1.html#T13" target="_self">Th13</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun3.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#T38">SEQM_3:38</a></i>
;<br/>
<b>hence </b><a NAME="E4:69"/>
<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="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <span class="p3">(<span class="default"><a href="partfun3.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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="ref" href="rinfsup1.html#T38" target="_self">Th38</a></i>;<br/>


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