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<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> , <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:67"/><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c2">f1</font> <a href="cfcont_1.html#R2" title="CFCONT_1:pred.2">is_continuous_on</a> <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E2:67"/><i><font color="Green" title="E46">A2</font></i>: 
<font color="Maroon" title="c3">f2</font> <a href="cfcont_1.html#R2" title="CFCONT_1:pred.2">is_continuous_on</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E3:67"/><i><font color="Green" title="E47">A3</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f1</font>
 
<b>by </b><i><a class="txt" href="cfcont_1.html#E1:67"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="cfcont_1.html#T60" target="_self" title="CFCONT_1:th.60">Th60</a></i>;<br/>
<a NAME="E4:67"/>
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f2</font>
 
<b>by </b><i><a class="txt" href="cfcont_1.html#E2:67"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="cfcont_1.html#T60" target="_self" title="CFCONT_1:th.60">Th60</a></i>;<br/>
<a NAME="E5:67"/><b>then </b><i><font color="Green" title="E48">A4</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="cfcont_1.html#E3:67"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E6:67"/><b>then </b><i><font color="Green" title="E49">A5</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c3">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="cfunct_1.html#T3" title="CFUNCT_1:th.3">CFUNCT_1:3</a></i>;<br/>
<a NAME="E7:67"/><i><font color="Green" title="E50">A6</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <font color="Maroon" title="c3">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="cfcont_1.html#E5:67"><i><font color="Green" title="E48">A4</font></i></a>, <a class="ref" href="cfunct_1.html#T4" title="CFUNCT_1:th.4">CFUNCT_1:4</a></i>;<br/>
<a NAME="E8:67"/><i><font color="Green" title="E51">A7</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="cfcont_1.html#E5:67"><i><font color="Green" title="E48">A4</font></i></a>, <a class="ref" href="cfunct_1.html#T5" title="CFUNCT_1:th.5">CFUNCT_1:5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">s1</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>;<br/><b>assume </b><a NAME="E1:67_1"/><i><font color="Green" title="E52">A8</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">s1</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="c4">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 ;<br/><a NAME="E2:67_1"/><b>then </b><i><font color="Green" title="E53">A9</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f2</font></span>)</span>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E5:67"><i><font color="Green" title="E48">A4</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E3:67_1"/><i><font color="Green" title="E54">A10</font></i>: 
( <font color="Maroon" title="c2">f1</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp; <font color="Maroon" title="c2">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:67"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="cfcont_1.html#E1:67_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="ref" href="cfcont_1.html#T60" target="_self" title="CFCONT_1:th.60">Th60</a></i>;<br/><a NAME="E4:67_1"/><i><font color="Green" title="E55">A11</font></i>: 
( <font color="Maroon" title="c3">f2</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp; <font color="Maroon" title="c3">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E2:67"><i><font color="Green" title="E46">A2</font></i></a>, <a class="txt" href="cfcont_1.html#E1:67_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="ref" href="cfcont_1.html#T60" target="_self" title="CFCONT_1:th.60">Th60</a></i>;<br/><a NAME="E5:67_1"/><b>then </b><i><font color="Green" title="E56">A12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E3:67_1"><i><font color="Green" title="E54">A10</font></i></a>, <a class="ref" href="comseq_2.html#T13" title="COMSEQ_2:th.13">COMSEQ_2:13</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s1</font></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="c2">f1</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <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">f2</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cfcont_1.html#E6:67"><i><font color="Green" title="E49">A5</font></i></a>, <a class="txt" href="cfcont_1.html#E1:67_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="txt" href="cfcont_1.html#E3:67_1"><i><font color="Green" title="E54">A10</font></i></a>, <a class="txt" href="cfcont_1.html#E4:67_1"><i><font color="Green" title="E55">A11</font></i></a>, <a class="ref" href="cfunct_1.html#T3" title="CFUNCT_1:th.3">CFUNCT_1:3</a></i>
<br/>.= 
 <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="c2">f1</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cfcont_1.html#E3:67_1"><i><font color="Green" title="E54">A10</font></i></a>, <a class="txt" href="cfcont_1.html#E4:67_1"><i><font color="Green" title="E55">A11</font></i></a>, <a class="ref" href="comseq_2.html#T14" title="COMSEQ_2:th.14">COMSEQ_2:14</a></i>
<br/>.= 
 <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="c2">f1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span>
<b>by </b><i><a class="txt" href="cfcont_1.html#E2:67_1"><i><font color="Green" title="E53">A9</font></i></a>, <a class="ref" href="cfcont_1.html#T18" target="_self" title="CFCONT_1:th.18">Th18</a></i>
;<br/><b>hence </b><a NAME="E7:67_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c2">f1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E2:67_1"><i><font color="Green" title="E53">A9</font></i></a>, <a class="txt" href="cfcont_1.html#E5:67_1"><i><font color="Green" title="E56">A12</font></i></a>, <a class="ref" href="cfcont_1.html#T18" target="_self" title="CFCONT_1:th.18">Th18</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:67"/>
<font color="Maroon" title="c2">f1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c3">f2</font> <a href="cfcont_1.html#R2" title="CFCONT_1:pred.2">is_continuous_on</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E6:67"><i><font color="Green" title="E49">A5</font></i></a>, <a class="ref" href="cfcont_1.html#T60" target="_self" title="CFCONT_1:th.60">Th60</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">s1</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>;<br/><b>assume </b><a NAME="E1:67_2"/><i><font color="Green" title="E52">A13</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">s1</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="c4">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 ;<br/><a NAME="E2:67_2"/><b>then </b><i><font color="Green" title="E53">A14</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f2</font></span>)</span>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E5:67"><i><font color="Green" title="E48">A4</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E3:67_2"/><i><font color="Green" title="E54">A15</font></i>: 
( <font color="Maroon" title="c2">f1</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp; <font color="Maroon" title="c2">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:67"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="cfcont_1.html#E1:67_2"><i><font color="Green" title="E52">A13</font></i></a>, <a class="ref" href="cfcont_1.html#T60" target="_self" title="CFCONT_1:th.60">Th60</a></i>;<br/><a NAME="E4:67_2"/><i><font color="Green" title="E55">A16</font></i>: 
( <font color="Maroon" title="c3">f2</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp; <font color="Maroon" title="c3">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E2:67"><i><font color="Green" title="E46">A2</font></i></a>, <a class="txt" href="cfcont_1.html#E1:67_2"><i><font color="Green" title="E52">A13</font></i></a>, <a class="ref" href="cfcont_1.html#T60" target="_self" title="CFCONT_1:th.60">Th60</a></i>;<br/><a NAME="E5:67_2"/><b>then </b><i><font color="Green" title="E56">A17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E3:67_2"><i><font color="Green" title="E54">A15</font></i></a>, <a class="ref" href="comseq_2.html#T25" title="COMSEQ_2:th.25">COMSEQ_2:25</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s1</font></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="c2">f1</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <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">f2</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cfcont_1.html#E7:67"><i><font color="Green" title="E50">A6</font></i></a>, <a class="txt" href="cfcont_1.html#E1:67_2"><i><font color="Green" title="E52">A13</font></i></a>, <a class="txt" href="cfcont_1.html#E3:67_2"><i><font color="Green" title="E54">A15</font></i></a>, <a class="txt" href="cfcont_1.html#E4:67_2"><i><font color="Green" title="E55">A16</font></i></a>, <a class="ref" href="cfunct_1.html#T4" title="CFUNCT_1:th.4">CFUNCT_1:4</a></i>
<br/>.= 
 <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="c2">f1</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cfcont_1.html#E3:67_2"><i><font color="Green" title="E54">A15</font></i></a>, <a class="txt" href="cfcont_1.html#E4:67_2"><i><font color="Green" title="E55">A16</font></i></a>, <a class="ref" href="comseq_2.html#T26" title="COMSEQ_2:th.26">COMSEQ_2:26</a></i>
<br/>.= 
 <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="c2">f1</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span>
<b>by </b><i><a class="txt" href="cfcont_1.html#E2:67_2"><i><font color="Green" title="E53">A14</font></i></a>, <a class="ref" href="cfcont_1.html#T18" target="_self" title="CFCONT_1:th.18">Th18</a></i>
;<br/><b>hence </b><a NAME="E7:67_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c2">f1</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E2:67_2"><i><font color="Green" title="E53">A14</font></i></a>, <a class="txt" href="cfcont_1.html#E5:67_2"><i><font color="Green" title="E56">A17</font></i></a>, <a class="ref" href="cfcont_1.html#T18" target="_self" title="CFCONT_1:th.18">Th18</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:67"/>
<font color="Maroon" title="c2">f1</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <font color="Maroon" title="c3">f2</font> <a href="cfcont_1.html#R2" title="CFCONT_1:pred.2">is_continuous_on</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E7:67"><i><font color="Green" title="E50">A6</font></i></a>, <a class="ref" href="cfcont_1.html#T60" target="_self" title="CFCONT_1:th.60">Th60</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">s1</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>;<br/><b>assume </b><a NAME="E1:67_3"/><i><font color="Green" title="E52">A18</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">s1</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="c4">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 ;<br/><a NAME="E2:67_3"/><b>then </b><i><font color="Green" title="E53">A19</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f2</font></span>)</span>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E5:67"><i><font color="Green" title="E48">A4</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E3:67_3"/><i><font color="Green" title="E54">A20</font></i>: 
( <font color="Maroon" title="c2">f1</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp; <font color="Maroon" title="c2">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:67"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="cfcont_1.html#E1:67_3"><i><font color="Green" title="E52">A18</font></i></a>, <a class="ref" href="cfcont_1.html#T60" target="_self" title="CFCONT_1:th.60">Th60</a></i>;<br/><a NAME="E4:67_3"/><i><font color="Green" title="E55">A21</font></i>: 
( <font color="Maroon" title="c3">f2</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp; <font color="Maroon" title="c3">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E2:67"><i><font color="Green" title="E46">A2</font></i></a>, <a class="txt" href="cfcont_1.html#E1:67_3"><i><font color="Green" title="E52">A18</font></i></a>, <a class="ref" href="cfcont_1.html#T60" target="_self" title="CFCONT_1:th.60">Th60</a></i>;<br/><a NAME="E5:67_3"/><b>then </b><i><font color="Green" title="E56">A22</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E3:67_3"><i><font color="Green" title="E54">A20</font></i></a>, <a class="ref" href="comseq_2.html#T29" title="COMSEQ_2:th.29">COMSEQ_2:29</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s1</font></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="c2">f1</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <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">f2</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cfcont_1.html#E8:67"><i><font color="Green" title="E51">A7</font></i></a>, <a class="txt" href="cfcont_1.html#E1:67_3"><i><font color="Green" title="E52">A18</font></i></a>, <a class="txt" href="cfcont_1.html#E3:67_3"><i><font color="Green" title="E54">A20</font></i></a>, <a class="txt" href="cfcont_1.html#E4:67_3"><i><font color="Green" title="E55">A21</font></i></a>, <a class="ref" href="cfunct_1.html#T5" title="CFUNCT_1:th.5">CFUNCT_1:5</a></i>
<br/>.= 
 <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="c2">f1</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cfcont_1.html#E3:67_3"><i><font color="Green" title="E54">A20</font></i></a>, <a class="txt" href="cfcont_1.html#E4:67_3"><i><font color="Green" title="E55">A21</font></i></a>, <a class="ref" href="comseq_2.html#T30" title="COMSEQ_2:th.30">COMSEQ_2:30</a></i>
<br/>.= 
 <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="c2">f1</font> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span>
<b>by </b><i><a class="txt" href="cfcont_1.html#E2:67_3"><i><font color="Green" title="E53">A19</font></i></a>, <a class="ref" href="cfcont_1.html#T18" target="_self" title="CFCONT_1:th.18">Th18</a></i>
;<br/><b>hence </b><a NAME="E7:67_3"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c2">f1</font> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E2:67_3"><i><font color="Green" title="E53">A19</font></i></a>, <a class="txt" href="cfcont_1.html#E5:67_3"><i><font color="Green" title="E56">A22</font></i></a>, <a class="ref" href="cfcont_1.html#T18" target="_self" title="CFCONT_1:th.18">Th18</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:67"/>
<font color="Maroon" title="c2">f1</font> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <font color="Maroon" title="c3">f2</font> <a href="cfcont_1.html#R2" title="CFCONT_1:pred.2">is_continuous_on</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E8:67"><i><font color="Green" title="E51">A7</font></i></a>, <a class="ref" href="cfcont_1.html#T60" target="_self" title="CFCONT_1:th.60">Th60</a></i>;<br/>


</div>
