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<span class="kw">let </span><font color="Maroon" title="c1">seq</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">seq</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>  &amp; <font color="Maroon" title="c1">seq</font> is  <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a>  implies 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">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c1">seq</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:63"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<font color="Maroon" title="c1">seq</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> 
 <span class="kw">and </span><br/><a NAME="E2:63"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c1">seq</font> is  <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c1">seq</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">m</font> be    <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> ; <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">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c1">seq</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">seq1</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font></span>)</span> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">m</font></span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">seq2</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><span class="kw"> such that </span><br/><a NAME="E5:63"/><span class="lab"><font color="Green" title="E30">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="c4">seq2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="seq_1.html#S1" title="SEQ_1:sch.1">SEQ_1:sch 1</a>();<br/></span>
<div><span class="lab"><font color="Green" title="E31">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">n</font> be    <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> ; <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="c3">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">seq2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font></span><br/><a NAME="E1:63_1"/>
( <font color="Maroon" title="c3">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c4">seq2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E2:63_1"/>
<font color="Maroon" title="c3">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">seq2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="seqm_3.html#T11" title="SEQM_3:th.11">SEQM_3:11</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/></div><span class="kw">end;</span></div>
<a NAME="E7:63"/>
<font color="Maroon" title="c3">seq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E8:63"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A5</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T40" target="_self" title="SEQ_4:th.40">Th40</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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="c4">seq2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">n</font>)</span><br/><a NAME="E1:63_2"/>
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E2:63_2"/>
<font color="Maroon" title="c4">seq2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63"><span class="lab"><font color="Green" title="E30">A3</font></span></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/></div><span class="kw">end;</span></div>
<a NAME="E10:63"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A6</font></span>: 
<font color="Maroon" title="c4">seq2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c1">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>;<br/>
<a NAME="E11:63"/><span class="kw">then </span>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">seq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c1">seq</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="seq_4.html#T30" target="_self" title="SEQ_4:th.30">Th30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:63"/>
<font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c1">seq</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E8:63"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E10:63"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E6:63"><span class="lab"><font color="Green" title="E31">A4</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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