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<span class="kw">let </span><font color="Maroon" title="c1">seq1</font>, <font color="Maroon" title="c2">seq2</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">seq1</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Maroon" title="c2">seq2</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c1">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">seq2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> ) implies (  <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c2">seq2</font> &amp;  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c2">seq2</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:100"/><span class="lab"><font color="Green" title="E77">A1</font></span>: 
<font color="Maroon" title="c1">seq1</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 <span class="kw">and </span><br/><a NAME="E2:100"/><span class="lab"><font color="Green" title="E78">A2</font></span>: 
<font color="Maroon" title="c2">seq2</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 <span class="kw">and </span><br/><a NAME="E3:100"/><span class="lab"><font color="Green" title="E79">A3</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c1">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">seq2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c2">seq2</font> &amp;  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c2">seq2</font> )</span><br/>

<a NAME="E4:100"/><span class="lab"><font color="Green" title="E80">A4</font></span>: 
(  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq1</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq1</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="ref" href="rinfsup1.html#T59" target="_self" title="RINFSUP1:th.59">Th59</a>, <a class="ref" href="rinfsup1.html#T60" target="_self" title="RINFSUP1:th.60">Th60</a></span>;<br/>
<a NAME="E5:100"/>
 <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c2">seq2</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:100"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="ref" href="rinfsup1.html#T58" target="_self" title="RINFSUP1:th.58">Th58</a></span>;<br/>
<a NAME="E6:100"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A5</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">inf</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:100"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="ref" href="rinfsup1.html#T25" target="_self" title="RINFSUP1:th.25">Th25</a>, <a class="ref" href="rinfsup1.html#T53" target="_self" title="RINFSUP1:th.53">Th53</a></span>;<br/>
<a NAME="E7:100"/>
 <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq1</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="ref" href="rinfsup1.html#T58" target="_self" title="RINFSUP1:th.58">Th58</a></span>;<br/>
<a NAME="E8:100"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A6</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="ref" href="rinfsup1.html#T24" target="_self" title="RINFSUP1:th.24">Th24</a>, <a class="ref" href="rinfsup1.html#T52" target="_self" title="RINFSUP1:th.52">Th52</a></span>;<br/>
<a NAME="E9:100"/>
 <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq1</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="ref" href="rinfsup1.html#T58" target="_self" title="RINFSUP1:th.58">Th58</a></span>;<br/>
<a NAME="E10:100"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A7</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">inf</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="ref" href="rinfsup1.html#T25" target="_self" title="RINFSUP1:th.25">Th25</a>, <a class="ref" href="rinfsup1.html#T53" target="_self" title="RINFSUP1:th.53">Th53</a></span>;<br/>
<a NAME="E11:100"/><span class="lab"><font color="Green" title="E84">A8</font></span>: 
(  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c2">seq2</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">seq2</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:100"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="ref" href="rinfsup1.html#T59" target="_self" title="RINFSUP1:th.59">Th59</a>, <a class="ref" href="rinfsup1.html#T60" target="_self" title="RINFSUP1:th.60">Th60</a></span>;<br/>
<a NAME="E12:100"/>
 <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">seq2</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:100"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="ref" href="rinfsup1.html#T58" target="_self" title="RINFSUP1:th.58">Th58</a></span>;<br/>
<a NAME="E13:100"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A9</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:100"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="ref" href="rinfsup1.html#T24" target="_self" title="RINFSUP1:th.24">Th24</a>, <a class="ref" href="rinfsup1.html#T52" target="_self" title="RINFSUP1:th.52">Th52</a></span>;<br/>
<a NAME="E14:100"/>
( ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> ) &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="txt" href="#E2:100"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="txt" href="#E3:100"><span class="lab"><font color="Green" title="E79">A3</font></span></a>, <a class="ref" href="rinfsup1.html#T77" target="_self" title="RINFSUP1:th.77">Th77</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:100"/>
(  <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c2">seq2</font> &amp;  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c2">seq2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:100"><span class="lab"><font color="Green" title="E80">A4</font></span></a>, <a class="txt" href="#E11:100"><span class="lab"><font color="Green" title="E84">A8</font></span></a>, <a class="txt" href="#E10:100"><span class="lab"><font color="Green" title="E83">A7</font></span></a>, <a class="txt" href="#E8:100"><span class="lab"><font color="Green" title="E82">A6</font></span></a>, <a class="txt" href="#E6:100"><span class="lab"><font color="Green" title="E81">A5</font></span></a>, <a class="txt" href="#E13:100"><span class="lab"><font color="Green" title="E85">A9</font></span></a>, <a class="ref" href="seq_2.html#T32" title="SEQ_2:th.32">SEQ_2:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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