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<b>let </b><font color="Maroon" title="c1">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:102"/><i><font color="Green" title="E83">A1</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 ;<br/>

<a NAME="E2:102"/><b>then </b><i><font color="Green" title="E84">A2</font></i>: 
( <font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> &amp; <font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> &amp;  <a href="partfun3.html#K6" title="PARTFUN3:func.6">-</a> <font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> )
 
<b>by </b><i><a class="ref" href="seq_2.html#D5" title="SEQ_2:def.5">SEQ_2:def 5</a>, <a class="ref" href="seqm_3.html#T70" title="SEQM_3:th.70">SEQM_3:70</a></i>;<br/>
<a NAME="E3:102"/>
(  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp;  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> )
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:102"><i><font color="Green" title="E83">A1</font></i></a>, <a class="ref" href="rinfsup1.html#T58" target="_self" title="RINFSUP1:th.58">Th58</a></i>;<br/>
<a NAME="E4:102"/><b>then </b><i><font color="Green" title="E85">A3</font></i>: 
(  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> &amp;  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> &amp;  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> &amp;  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="seq_2.html#D5" title="SEQ_2:def.5">SEQ_2:def 5</a></i>;<br/>
<i><font color="Green" title="E86">A4</font></i>:  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c1">seq</font></span>)</span> = 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K2" title="RINFSUP1:func.2">inf</a> <span class="p3">(<span class="default"><a href="partfun3.html#K6" title="PARTFUN3:func.6">-</a> <span class="p4">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rinfsup1.html#E4:102"><i><font color="Green" title="E85">A3</font></i></a>, <a class="ref" href="rinfsup1.html#T13" target="_self" title="RINFSUP1:th.13">Th13</a></i>
<br/>.= 
 <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">inf</a> <span class="p1">(<span class="default"><a href="partfun3.html#K6" title="PARTFUN3:func.6">-</a> <span class="p2">(<span class="default"><a href="partfun3.html#K6" title="PARTFUN3:func.6">-</a> <span class="p3">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <span class="p4">(<span class="default"><a href="partfun3.html#K6" title="PARTFUN3:func.6">-</a> <font color="Maroon" title="c1">seq</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rinfsup1.html#E2:102"><i><font color="Green" title="E84">A2</font></i></a>, <a class="ref" href="rinfsup1.html#T63" target="_self" title="RINFSUP1:th.63">Th63</a></i>
<br/>.= 
 <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <span class="p1">(<span class="default"><a href="partfun3.html#K6" title="PARTFUN3:func.6">-</a> <font color="Maroon" title="c1">seq</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T36" title="SEQ_1:th.36">SEQ_1:36</a></i>
;<br/>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c1">seq</font></span>)</span> = 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p3">(<span class="default"><a href="partfun3.html#K6" title="PARTFUN3:func.6">-</a> <span class="p4">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rinfsup1.html#E4:102"><i><font color="Green" title="E85">A3</font></i></a>, <a class="ref" href="rinfsup1.html#T14" target="_self" title="RINFSUP1:th.14">Th14</a></i>
<br/>.= 
 <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><a href="partfun3.html#K6" title="PARTFUN3:func.6">-</a> <span class="p2">(<span class="default"><a href="partfun3.html#K6" title="PARTFUN3:func.6">-</a> <span class="p3">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <span class="p4">(<span class="default"><a href="partfun3.html#K6" title="PARTFUN3:func.6">-</a> <font color="Maroon" title="c1">seq</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rinfsup1.html#E2:102"><i><font color="Green" title="E84">A2</font></i></a>, <a class="ref" href="rinfsup1.html#T64" target="_self" title="RINFSUP1:th.64">Th64</a></i>
<br/>.= 
 <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <span class="p1">(<span class="default"><a href="partfun3.html#K6" title="PARTFUN3:func.6">-</a> <font color="Maroon" title="c1">seq</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T36" title="SEQ_1:th.36">SEQ_1:36</a></i>
;<br/>
<b>hence </b><a NAME="E7:102"/>
(  <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <span class="p1">(<span class="default"><a href="partfun3.html#K6" title="PARTFUN3:func.6">-</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c1">seq</font></span>)</span> &amp;  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <span class="p1">(<span class="default"><a href="partfun3.html#K6" title="PARTFUN3:func.6">-</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c1">seq</font></span>)</span> )
 <b>by </b><i><a class="txt" href="rinfsup1.html#E5:102"><i><font color="Green" title="E86">A4</font></i></a></i>;<br/>


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