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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="E1:87"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V2">bounded_below</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V2">bounded_below</a> )
 ;<br/>

<a NAME="E2:87"/><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#V2">bounded_below</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#V2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E3:87"/><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">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#K3">+</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#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">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#K3">+</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">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#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E4:87"/><b>then </b><i><font color="Green">E75</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">c<sub>2</sub></font> <a href="partfun3.html#K3">+</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#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">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#K3">+</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">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#T37">SEQM_3:37</a></i>;<br/>
<a NAME="E5:87"/>
( <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_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#K2" target="_self">inf</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#K3" target="_self">inferior_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#K2" target="_self">inf</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#T38" target="_self">Th38</a></i>;<br/>
<b>hence </b><a NAME="E6:87"/>
<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">c<sub>2</sub></font> <a href="partfun3.html#K3">+</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#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">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#K3">+</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">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#E4:87"><i><font color="Green">E75</font></i></a>, <a class="ref" href="rinfsup1.html#T38" target="_self">Th38</a></i>;<br/>


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