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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">seq</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">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="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Maroon" title="c1">seq</font> &amp; <font color="Olive" title="b1">seq1</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:47"/>
( <font color="Maroon" title="c1">seq</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Maroon" title="c1">seq</font> &amp; <font color="Olive" title="b1">seq1</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:47"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
<font color="Maroon" title="c1">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c2">k</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cfcont_1.html#T46" target="_self" title="CFCONT_1:th.46">Th46</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c1">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c2">k</font>
; <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="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c2">k</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Maroon" title="c1">seq</font> &amp; <font color="Maroon" title="c1">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c2">k</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E4:47"/>
( <font color="Maroon" title="c1">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c2">k</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Maroon" title="c1">seq</font> &amp; <font color="Maroon" title="c1">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c2">k</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E15">A1</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/>


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