<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">s1</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="c3">s</font> is <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> &amp; <font color="Maroon" title="c4">s1</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  implies  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  )</span><br/>



<b>assume </b><a NAME="E1:56"/><i><font color="Green" title="E29">A1</font></i>: 
( <font color="Maroon" title="c3">s</font> is <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> &amp; <font color="Maroon" title="c4">s1</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> </span><br/>

<a NAME="E2:56"/><b>then </b>
<font color="Maroon" title="c3">s</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c4">s1</font> is <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>
 
<b>by </b><i><a class="ref" href="comseq_2.html#T42" target="_self" title="COMSEQ_2:th.42">Th42</a></i>;<br/>
<b>hence </b> <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<b>by </b><i><a class="ref" href="comseq_2.html#T12" target="_self" title="COMSEQ_2:th.12">Th12</a></i>
<br/>.= 
 <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
<b>by </b><i><a class="txt" href="#E1:56"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="comseq_2.html#T43" target="_self" title="COMSEQ_2:th.43">Th43</a>, <a class="ref" href="complex1.html#T113" title="COMPLEX1:th.113">COMPLEX1:113</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
