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<b>let </b><font color="Maroon" title="c1">seq</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">seq</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  ex <font color="Olive" title="b1">seq1</font> being  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> st <br/>( <font color="Olive" title="b1">seq1</font> is    <a href="comseq_3.html#M1" title="COMSEQ_3:mode.1">subsequence</a> of <font color="Maroon" title="c1">seq</font> &amp; <font color="Olive" title="b1">seq1</font> is <a href="comseq_1.html#V1" title="COMSEQ_1:attr.1">non-zero</a> ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green" title="E34">A2</font></i>: 
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">seq1</font> being  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> st <br/>( <font color="Olive" title="b1">seq1</font> is    <a href="comseq_3.html#M1" title="COMSEQ_3:mode.1">subsequence</a> of <font color="Maroon" title="c1">seq</font> &amp; <font color="Olive" title="b1">seq1</font> is <a href="comseq_1.html#V1" title="COMSEQ_1:attr.1">non-zero</a> )</span><br/>

<b>consider </b><font color="Maroon" title="c2">k</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> <b> such that </b><br/><a NAME="E4:48"/><i><font color="Green" title="E35">A3</font></i>: 
<font color="Maroon" title="c1">seq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> <font color="Maroon" title="c2">k</font> is <a href="comseq_1.html#V1" title="COMSEQ_1:attr.1">non-zero</a>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:48"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="cfcont_1.html#E2:48"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="cfcont_1.html#T46" target="_self" title="CFCONT_1:th.46">Th46</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c1">seq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> <font color="Maroon" title="c2">k</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">seq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> <font color="Maroon" title="c2">k</font> is    <a href="comseq_3.html#M1" title="COMSEQ_3:mode.1">subsequence</a> of <font color="Maroon" title="c1">seq</font> &amp; <font color="Maroon" title="c1">seq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> <font color="Maroon" title="c2">k</font> is <a href="comseq_1.html#V1" title="COMSEQ_1:attr.1">non-zero</a> )</span><br/>

<b>thus </b><a NAME="E5:48"/>
( <font color="Maroon" title="c1">seq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> <font color="Maroon" title="c2">k</font> is    <a href="comseq_3.html#M1" title="COMSEQ_3:mode.1">subsequence</a> of <font color="Maroon" title="c1">seq</font> &amp; <font color="Maroon" title="c1">seq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> <font color="Maroon" title="c2">k</font> is <a href="comseq_1.html#V1" title="COMSEQ_1:attr.1">non-zero</a> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E4:48"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="cfcont_1.html#T38" target="_self" title="CFCONT_1:th.38">Th38</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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