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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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">seq1</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp; <font color="Maroon" title="c2">seq2</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c1">seq1</font> <a href="funct_1.html#K1" title="FUNCT_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="funct_1.html#K1" title="FUNCT_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="E1:92"/><span class="lab"><font color="Green" title="E74">A1</font></span>: 
<font color="Maroon" title="c1">seq1</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 <span class="kw">and </span><br/><a NAME="E2:92"/><span class="lab"><font color="Green" title="E75">A2</font></span>: 
<font color="Maroon" title="c2">seq2</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 <span class="kw">and </span><br/><a NAME="E3:92"/><span class="lab"><font color="Green" title="E76">A3</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c1">seq1</font> <a href="funct_1.html#K1" title="FUNCT_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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E4:92"/><span class="lab"><font color="Green" title="E77">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="comseq_2.html#V2" title="COMSEQ_2:attr.2">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="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="ref" href="rinfsup1.html#T57" target="_self" onmouseover="rs('rinfsup1/T57')" onmouseout="rh()">Th57</a>, <a class="ref" href="rinfsup1.html#T58" target="_self" onmouseover="rs('rinfsup1/T58')" onmouseout="rh()">Th58</a></span>;<br/>
<a NAME="E5:92"/>
 <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c2">seq2</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E75">A2</font></span></a>, <a class="ref" href="rinfsup1.html#T56" target="_self" onmouseover="rs('rinfsup1/T56')" onmouseout="rh()">Th56</a></span>;<br/>
<a NAME="E6:92"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A5</font></span>: 
 <a href="seq_2.html#K1" title="SEQ_2:func.1">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">lower_bound</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="#E2:92"><span class="lab"><font color="Green" title="E75">A2</font></span></a>, <a class="ref" href="rinfsup1.html#T25" target="_self" onmouseover="rs('rinfsup1/T25')" onmouseout="rh()">Th25</a>, <a class="ref" href="rinfsup1.html#T51" target="_self" onmouseover="rs('rinfsup1/T51')" onmouseout="rh()">Th51</a></span>;<br/>
<a NAME="E7:92"/>
 <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq1</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="ref" href="rinfsup1.html#T56" target="_self" onmouseover="rs('rinfsup1/T56')" onmouseout="rh()">Th56</a></span>;<br/>
<a NAME="E8:92"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A6</font></span>: 
 <a href="seq_2.html#K1" title="SEQ_2:func.1">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">upper_bound</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="#E1:92"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="ref" href="rinfsup1.html#T24" target="_self" onmouseover="rs('rinfsup1/T24')" onmouseout="rh()">Th24</a>, <a class="ref" href="rinfsup1.html#T50" target="_self" onmouseover="rs('rinfsup1/T50')" onmouseout="rh()">Th50</a></span>;<br/>
<a NAME="E9:92"/>
 <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq1</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="ref" href="rinfsup1.html#T56" target="_self" onmouseover="rs('rinfsup1/T56')" onmouseout="rh()">Th56</a></span>;<br/>
<a NAME="E10:92"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A7</font></span>: 
 <a href="seq_2.html#K1" title="SEQ_2:func.1">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">lower_bound</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="#E1:92"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="ref" href="rinfsup1.html#T25" target="_self" onmouseover="rs('rinfsup1/T25')" onmouseout="rh()">Th25</a>, <a class="ref" href="rinfsup1.html#T51" target="_self" onmouseover="rs('rinfsup1/T51')" onmouseout="rh()">Th51</a></span>;<br/>
<a NAME="E11:92"/><span class="lab"><font color="Green" title="E81">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="comseq_2.html#V2" title="COMSEQ_2:attr.2">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="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E75">A2</font></span></a>, <a class="ref" href="rinfsup1.html#T57" target="_self" onmouseover="rs('rinfsup1/T57')" onmouseout="rh()">Th57</a>, <a class="ref" href="rinfsup1.html#T58" target="_self" onmouseover="rs('rinfsup1/T58')" onmouseout="rh()">Th58</a></span>;<br/>
<a NAME="E12:92"/>
 <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">seq2</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E75">A2</font></span></a>, <a class="ref" href="rinfsup1.html#T56" target="_self" onmouseover="rs('rinfsup1/T56')" onmouseout="rh()">Th56</a></span>;<br/>
<a NAME="E13:92"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A9</font></span>: 
 <a href="seq_2.html#K1" title="SEQ_2:func.1">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">upper_bound</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="#E2:92"><span class="lab"><font color="Green" title="E75">A2</font></span></a>, <a class="ref" href="rinfsup1.html#T24" target="_self" onmouseover="rs('rinfsup1/T24')" onmouseout="rh()">Th24</a>, <a class="ref" href="rinfsup1.html#T50" target="_self" onmouseover="rs('rinfsup1/T50')" onmouseout="rh()">Th50</a></span>;<br/>
<a NAME="E14:92"/>
( (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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="funct_1.html#K1" title="FUNCT_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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> ) &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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="funct_1.html#K1" title="FUNCT_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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E75">A2</font></span></a>, <a class="txt" href="#E3:92"><span class="lab"><font color="Green" title="E76">A3</font></span></a>, <a class="ref" href="rinfsup1.html#T75" target="_self" onmouseover="rs('rinfsup1/T75')" onmouseout="rh()">Th75</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:92"/>
(  <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="#E4:92"><span class="lab"><font color="Green" title="E77">A4</font></span></a>, <a class="txt" href="#E11:92"><span class="lab"><font color="Green" title="E81">A8</font></span></a>, <a class="txt" href="#E10:92"><span class="lab"><font color="Green" title="E80">A7</font></span></a>, <a class="txt" href="#E8:92"><span class="lab"><font color="Green" title="E79">A6</font></span></a>, <a class="txt" href="#E6:92"><span class="lab"><font color="Green" title="E78">A5</font></span></a>, <a class="txt" href="#E13:92"><span class="lab"><font color="Green" title="E82">A9</font></span></a>, <a class="ref" href="seq_2.html#T18" onmouseover="rs('seq_2/T18')" onmouseout="rh()">SEQ_2:18</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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