<?xml version="1.0"?>
<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#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:19"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c2">f1</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E2:19"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c3">f2</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E3:19"/><i><font color="Green" title="E17">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="fcont_1.html#E1:19"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="fcont_1.html#T14" target="_self" title="FCONT_1:th.14">Th14</a></i>;<br/>
<a NAME="E4:19"/>
<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="fcont_1.html#E2:19"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="fcont_1.html#T14" target="_self" title="FCONT_1:th.14">Th14</a></i>;<br/>
<a NAME="E5:19"/><b>then </b><i><font color="Green" title="E18">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="fcont_1.html#E3:19"><i><font color="Green" title="E17">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="E6:19"/><b>then </b><i><font color="Green" title="E19">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="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c3">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D3" title="SEQ_1:def.3">SEQ_1:def 3</a></i>;<br/>
<a NAME="E7:19"/><i><font color="Green" title="E20">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="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c3">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="fcont_1.html#E5:19"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>;<br/>
<a NAME="E8:19"/><i><font color="Green" title="E21">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="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="fcont_1.html#E5:19"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="seq_1.html#D5" title="SEQ_1:def.5">SEQ_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">s1</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:19_1"/><i><font color="Green" title="E22">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="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_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="E2:19_1"/><b>then </b><i><font color="Green" title="E23">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="fcont_1.html#E5:19"><i><font color="Green" title="E18">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="E3:19_1"/><i><font color="Green" title="E24">A10</font></i>: 
( <font color="Maroon" title="c2">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_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="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="fcont_1.html#E1:19"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="fcont_1.html#E1:19_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="fcont_1.html#T14" target="_self" title="FCONT_1:th.14">Th14</a></i>;<br/><a NAME="E4:19_1"/><i><font color="Green" title="E25">A11</font></i>: 
( <font color="Maroon" title="c3">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c3">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_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="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="fcont_1.html#E2:19"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="fcont_1.html#E1:19_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="fcont_1.html#T14" target="_self" title="FCONT_1:th.14">Th14</a></i>;<br/><a NAME="E5:19_1"/><b>then </b><i><font color="Green" title="E26">A12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="fcont_1.html#E3:19_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="seq_2.html#T19" title="SEQ_2:th.19">SEQ_2:19</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fcont_1.html#E6:19"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="fcont_1.html#E1:19_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="fcont_1.html#E3:19_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="fcont_1.html#E4:19_1"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="seq_1.html#D3" title="SEQ_1:def.3">SEQ_1:def 3</a></i>
<br/>.= 
 <a href="seq_2.html#K2" title="SEQ_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="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fcont_1.html#E3:19_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="fcont_1.html#E4:19_1"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="seq_2.html#T20" title="SEQ_2:th.20">SEQ_2:20</a></i>
<br/>.= 
 <a href="seq_2.html#K2" title="SEQ_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="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span>
<b>by </b><i><a class="txt" href="fcont_1.html#E2:19_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="rfunct_2.html#T23" title="RFUNCT_2:th.23">RFUNCT_2:23</a></i>
;<br/><b>hence </b><a NAME="E7:19_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_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="seq_2.html#K2" title="SEQ_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="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="fcont_1.html#E2:19_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="fcont_1.html#E5:19_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="rfunct_2.html#T23" title="RFUNCT_2:th.23">RFUNCT_2:23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:19"/>
<font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c3">f2</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="fcont_1.html#E6:19"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="fcont_1.html#T14" target="_self" title="FCONT_1:th.14">Th14</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">s1</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:19_2"/><i><font color="Green" title="E22">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="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_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="E2:19_2"/><b>then </b><i><font color="Green" title="E23">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="fcont_1.html#E5:19"><i><font color="Green" title="E18">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="E3:19_2"/><i><font color="Green" title="E24">A15</font></i>: 
( <font color="Maroon" title="c2">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_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="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="fcont_1.html#E1:19"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="fcont_1.html#E1:19_2"><i><font color="Green" title="E22">A13</font></i></a>, <a class="ref" href="fcont_1.html#T14" target="_self" title="FCONT_1:th.14">Th14</a></i>;<br/><a NAME="E4:19_2"/><i><font color="Green" title="E25">A16</font></i>: 
( <font color="Maroon" title="c3">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c3">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_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="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="fcont_1.html#E2:19"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="fcont_1.html#E1:19_2"><i><font color="Green" title="E22">A13</font></i></a>, <a class="ref" href="fcont_1.html#T14" target="_self" title="FCONT_1:th.14">Th14</a></i>;<br/><a NAME="E5:19_2"/><b>then </b><i><font color="Green" title="E26">A17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="fcont_1.html#E3:19_2"><i><font color="Green" title="E24">A15</font></i></a>, <a class="ref" href="seq_2.html#T25" title="SEQ_2:th.25">SEQ_2:25</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fcont_1.html#E7:19"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="fcont_1.html#E1:19_2"><i><font color="Green" title="E22">A13</font></i></a>, <a class="txt" href="fcont_1.html#E3:19_2"><i><font color="Green" title="E24">A15</font></i></a>, <a class="txt" href="fcont_1.html#E4:19_2"><i><font color="Green" title="E25">A16</font></i></a>, <a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>
<br/>.= 
 <a href="seq_2.html#K2" title="SEQ_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="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fcont_1.html#E3:19_2"><i><font color="Green" title="E24">A15</font></i></a>, <a class="txt" href="fcont_1.html#E4:19_2"><i><font color="Green" title="E25">A16</font></i></a>, <a class="ref" href="seq_2.html#T26" title="SEQ_2:th.26">SEQ_2:26</a></i>
<br/>.= 
 <a href="seq_2.html#K2" title="SEQ_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="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span>
<b>by </b><i><a class="txt" href="fcont_1.html#E2:19_2"><i><font color="Green" title="E23">A14</font></i></a>, <a class="ref" href="rfunct_2.html#T23" title="RFUNCT_2:th.23">RFUNCT_2:23</a></i>
;<br/><b>hence </b><a NAME="E7:19_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_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="seq_2.html#K2" title="SEQ_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="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="fcont_1.html#E2:19_2"><i><font color="Green" title="E23">A14</font></i></a>, <a class="txt" href="fcont_1.html#E5:19_2"><i><font color="Green" title="E26">A17</font></i></a>, <a class="ref" href="rfunct_2.html#T23" title="RFUNCT_2:th.23">RFUNCT_2:23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:19"/>
<font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c3">f2</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="fcont_1.html#E7:19"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="fcont_1.html#T14" target="_self" title="FCONT_1:th.14">Th14</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">s1</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:19_3"/><i><font color="Green" title="E22">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="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_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="E2:19_3"/><b>then </b><i><font color="Green" title="E23">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="fcont_1.html#E5:19"><i><font color="Green" title="E18">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="E3:19_3"/><i><font color="Green" title="E24">A20</font></i>: 
( <font color="Maroon" title="c2">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_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="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="fcont_1.html#E1:19"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="fcont_1.html#E1:19_3"><i><font color="Green" title="E22">A18</font></i></a>, <a class="ref" href="fcont_1.html#T14" target="_self" title="FCONT_1:th.14">Th14</a></i>;<br/><a NAME="E4:19_3"/><i><font color="Green" title="E25">A21</font></i>: 
( <font color="Maroon" title="c3">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c3">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_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="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="fcont_1.html#E2:19"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="fcont_1.html#E1:19_3"><i><font color="Green" title="E22">A18</font></i></a>, <a class="ref" href="fcont_1.html#T14" target="_self" title="FCONT_1:th.14">Th14</a></i>;<br/><a NAME="E5:19_3"/><b>then </b><i><font color="Green" title="E26">A22</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="fcont_1.html#E3:19_3"><i><font color="Green" title="E24">A20</font></i></a>, <a class="ref" href="seq_2.html#T28" title="SEQ_2:th.28">SEQ_2:28</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fcont_1.html#E8:19"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="fcont_1.html#E1:19_3"><i><font color="Green" title="E22">A18</font></i></a>, <a class="txt" href="fcont_1.html#E3:19_3"><i><font color="Green" title="E24">A20</font></i></a>, <a class="txt" href="fcont_1.html#E4:19_3"><i><font color="Green" title="E25">A21</font></i></a>, <a class="ref" href="seq_1.html#D5" title="SEQ_1:def.5">SEQ_1:def 5</a></i>
<br/>.= 
 <a href="seq_2.html#K2" title="SEQ_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="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fcont_1.html#E3:19_3"><i><font color="Green" title="E24">A20</font></i></a>, <a class="txt" href="fcont_1.html#E4:19_3"><i><font color="Green" title="E25">A21</font></i></a>, <a class="ref" href="seq_2.html#T29" title="SEQ_2:th.29">SEQ_2:29</a></i>
<br/>.= 
 <a href="seq_2.html#K2" title="SEQ_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="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span>
<b>by </b><i><a class="txt" href="fcont_1.html#E2:19_3"><i><font color="Green" title="E23">A19</font></i></a>, <a class="ref" href="rfunct_2.html#T23" title="RFUNCT_2:th.23">RFUNCT_2:23</a></i>
;<br/><b>hence </b><a NAME="E7:19_3"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_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="seq_2.html#K2" title="SEQ_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="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c4">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="fcont_1.html#E2:19_3"><i><font color="Green" title="E23">A19</font></i></a>, <a class="txt" href="fcont_1.html#E5:19_3"><i><font color="Green" title="E26">A22</font></i></a>, <a class="ref" href="rfunct_2.html#T23" title="RFUNCT_2:th.23">RFUNCT_2:23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:19"/>
<font color="Maroon" title="c2">f1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <font color="Maroon" title="c3">f2</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="fcont_1.html#E8:19"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="fcont_1.html#T14" target="_self" title="FCONT_1:th.14">Th14</a></i>;<br/>


</div>
