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<div class="add">

<b>let </b><font color="Maroon">seq</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:65"/>
<font color="Maroon">seq</font> is <a href="comseq_2.html#V1">bounded</a>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:65"/><i><font color="Green">E35</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> ) )
 <b>by </b><i><a class="ref" href="comseq_2.html#T8">COMSEQ_2:8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:65_1"/><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/><a NAME="E2:65_1"/><i><font color="Green">E38</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:65_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:65_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="comseq_3.html#E6"><i><font color="Green">Lemma42</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:65"/><b>then </b>
 <a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i>, <a class="ref" href="seq_2.html#T15">SEQ_2:15</a></i>;<br/>
<b>then consider </b><font color="Maroon">rseq1</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E7:65"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">rseq1</font> is    <a href="seqm_3.html#M1">subsequence</a> of  <a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">seq</font> &amp; <font color="Maroon">rseq1</font> is <a href="seq_2.html#V4">convergent</a> )
 <b>by </b><i><a class="ref" href="seq_4.html#T57">SEQ_4:57</a></i>;<br/>
<b>consider </b><font color="Maroon">Nseq</font> being  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E9:65"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">rseq1</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">seq</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Nseq</font>
 <b>by </b><i><a class="txt" href="comseq_3.html#E6"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/>
<div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:65_2"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">Nseq</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/><b>hence </b><a NAME="E2:65_2"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:65_3"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="comseq_3.html#E9:65"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E2:65_3"/><i><font color="Green">E47</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:65_3"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:65_3"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="comseq_3.html#E10:65"><i><font color="Green">E45</font></i></a>, <a class="txt" href="comseq_3.html#E1:65_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:65"/><b>then </b>
 <a href="comseq_3.html#K4" target="_self">Im</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq</font></span>)</span> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i>, <a class="ref" href="seq_2.html#T15">SEQ_2:15</a></i>;<br/>
<b>then consider </b><font color="Maroon">rseq2</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E14:65"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">rseq2</font> is    <a href="seqm_3.html#M1">subsequence</a> of  <a href="comseq_3.html#K4" target="_self">Im</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq</font></span>)</span> &amp; <font color="Maroon">rseq2</font> is <a href="seq_2.html#V4">convergent</a> )
 <b>by </b><i><a class="ref" href="seq_4.html#T57">SEQ_4:57</a></i>;<br/>
<b>consider </b><font color="Maroon">Nseq1</font> being  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E16:65"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">rseq2</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq</font></span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Nseq1</font>
 <b>by </b><i><a class="txt" href="comseq_3.html#E10:65"><i><font color="Green">E45</font></i></a>, <a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/>
<a NAME="E17:65"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">seq</font> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq</font> is    <a href="comseq_3.html#M1" target="_self">subsequence</a> of <font color="Maroon">seq</font>
 
<b>by </b><i><a class="txt" href="comseq_3.html#E7:65"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E18:65"/>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq</font></span>)</span> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq1</font> is    <a href="comseq_3.html#M1" target="_self">subsequence</a> of <font color="Maroon">seq</font> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq</font>
 
<b>by </b><i><a class="txt" href="comseq_3.html#E7:65"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E19:65"/><b>then </b><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq</font></span>)</span> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq1</font> is    <a href="comseq_3.html#M1" target="_self">subsequence</a> of <font color="Maroon">seq</font>
 
<b>by </b><i>, <a class="ref" href="comseq_3.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E20:65"/><i><font color="Green">E84</font></i>: 
 <a href="comseq_3.html#K4" target="_self">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq</font></span>)</span> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq1</font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="comseq_3.html#E10:65"><i><font color="Green">E45</font></i></a>, , </i>;<br/>
 <a href="comseq_3.html#K3" target="_self">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq</font></span>)</span> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq</font></span>)</span></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Nseq1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">rseq1</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Nseq1</font>
<b>by </b><i>, </i>
;<br/>
<a NAME="E22:65"/><b>then </b>
 <a href="comseq_3.html#K3" target="_self">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq</font></span>)</span> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq1</font></span>)</span> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">rseq1</font>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/>
<a NAME="E23:65"/><b>then </b>
 <a href="comseq_3.html#K3" target="_self">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq</font></span>)</span> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq1</font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="comseq_3.html#E6"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="seq_4.html#T29">SEQ_4:29</a></i>;<br/>
<a NAME="E24:65"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq</font></span>)</span> <a href="comseq_3.html#K5" target="_self">*</a> <font color="Maroon">Nseq1</font> is <a href="comseq_2.html#V2">convergent</a>
 
<b>by </b><i><a class="txt" href="comseq_3.html#E16:65"><i><font color="Green">E49</font></i></a>, <a class="txt" href="comseq_3.html#E3:65"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E25:65"/>
 ex <font color="Olive">seq1</font> being  <a href="comseq_1.html#NM1">Complex_Sequence</a> st <br/>( <font color="Olive">seq1</font> is    <a href="comseq_3.html#M1" target="_self">subsequence</a> of <font color="Maroon">seq</font> &amp; <font color="Olive">seq1</font> is <a href="comseq_2.html#V2">convergent</a> )
 <b>by </b><i><a class="txt" href="comseq_3.html#E14:65"><i><font color="Green">E48</font></i></a></i>;<br/>


</div>
