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

<b>let </b><font color="Maroon">CNS1</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">CNS2</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f1</font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">CNS1</font>,<font color="Maroon">CNS2</font>, <font color="Maroon">f2</font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">CNS1</font>,<font color="Maroon">CNS2</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:109"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f1</font> <a href="ncfcont1.html#R7" target="_self">is_continuous_on</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E2:109"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f2</font> <a href="ncfcont1.html#R7" target="_self">is_continuous_on</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E3:109"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i><a class="ref" href="ncfcont1.html#T16" target="_self">Th16</a>, <a class="ref" href="ncfcont1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E4:109"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font>
 
<b>by </b><i><a class="ref" href="ncfcont1.html#T17" target="_self">Th17</a>, <a class="ref" href="ncfcont1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E5:109"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E6:109"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_2.html#K1">+</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vfunct_2.html#D1">VFUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:109"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_2.html#K2">-</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ncfcont1.html#T18" target="_self">Th18</a>, <a class="ref" href="vfunct_2.html#D2">VFUNCT_2:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s1</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">CNS1</font>;<br/><b>assume </b><a NAME="E1:109_1"/><i><font color="Green">E112</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">s1</font> is <a href="clvect_1.html#V6">convergent</a> &amp;  <a href="clvect_1.html#K11">lim</a> <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/><a NAME="E2:109_1"/><b>then </b><i><font color="Green">E113</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="ncfcont1.html#T18" target="_self">Th18</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E3:109_1"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">f1</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font> is <a href="clvect_1.html#V6">convergent</a> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="clvect_1.html#K11">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="ncfcont1.html#T16" target="_self">Th16</a>, <a class="ref" href="ncfcont1.html#T20" target="_self">Th20</a>, <a class="ref" href="ncfcont1.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E4:109_1"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">f2</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font> is <a href="clvect_1.html#V6">convergent</a> &amp; <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="clvect_1.html#K11">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="ncfcont1.html#T17" target="_self">Th17</a>, <a class="ref" href="ncfcont1.html#T20" target="_self">Th20</a>, <a class="ref" href="ncfcont1.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E5:109_1"/><b>then </b><i><font color="Green">E116</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> is <a href="clvect_1.html#V6">convergent</a>
 <b>by </b><i><a class="ref" href="ncfcont1.html#T22" target="_self">Th22</a>, <a class="ref" href="clvect_1.html#T115">CLVECT_1:115</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_2.html#K1">+</a> <font color="Maroon">f2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">s1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="ncfcont1.html#T20" target="_self">Th20</a>, <a class="ref" href="ncfcont1.html#T22" target="_self">Th22</a>, , <a class="ref" href="vfunct_2.html#D1">VFUNCT_2:def 1</a></i>
<br/>.= 
 <a href="clvect_1.html#K11">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ncfcont1.html#T22" target="_self">Th22</a>, , <a class="ref" href="clvect_1.html#T121">CLVECT_1:121</a></i>
<br/>.= 
 <a href="clvect_1.html#K11">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_2.html#K1">+</a> <font color="Maroon">f2</font></span>)</span> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font></span>)</span>
<b>by </b><i><a class="ref" href="ncfcont1.html#T21" target="_self">Th21</a>, <a class="ref" href="ncfcont1.html#T4" target="_self">Th4</a></i>
;<br/><b>hence </b><a NAME="E7:109_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_2.html#K1">+</a> <font color="Maroon">f2</font></span>)</span> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font> is <a href="clvect_1.html#V6">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_2.html#K1">+</a> <font color="Maroon">f2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="clvect_1.html#K11">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_2.html#K1">+</a> <font color="Maroon">f2</font></span>)</span> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="ncfcont1.html#T21" target="_self">Th21</a>, , <a class="ref" href="ncfcont1.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:109"/>
<font color="Maroon">f1</font> <a href="vfunct_2.html#K1">+</a> <font color="Maroon">f2</font> <a href="ncfcont1.html#R7" target="_self">is_continuous_on</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="ncfcont1.html#T8" target="_self">Th8</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s1</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">CNS1</font>;<br/><b>assume </b><a NAME="E1:109_2"/><i><font color="Green">E118</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">s1</font> is <a href="clvect_1.html#V6">convergent</a> &amp;  <a href="clvect_1.html#K11">lim</a> <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/><a NAME="E2:109_2"/><b>then </b><i><font color="Green">E119</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="ncfcont1.html#T18" target="_self">Th18</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E3:109_2"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">f1</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font> is <a href="clvect_1.html#V6">convergent</a> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="clvect_1.html#K11">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="ncfcont1.html#T16" target="_self">Th16</a>, <a class="ref" href="ncfcont1.html#T20" target="_self">Th20</a>, <a class="ref" href="ncfcont1.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E4:109_2"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">f2</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font> is <a href="clvect_1.html#V6">convergent</a> &amp; <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="clvect_1.html#K11">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="ncfcont1.html#T17" target="_self">Th17</a>, <a class="ref" href="ncfcont1.html#T20" target="_self">Th20</a>, <a class="ref" href="ncfcont1.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E5:109_2"/><b>then </b><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> <a href="clvect_1.html#K7">-</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> is <a href="clvect_1.html#V6">convergent</a>
 <b>by </b><i><a class="ref" href="ncfcont1.html#T22" target="_self">Th22</a>, <a class="ref" href="clvect_1.html#T116">CLVECT_1:116</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_2.html#K2">-</a> <font color="Maroon">f2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">s1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ncfcont1.html#T19" target="_self">Th19</a>, <a class="ref" href="ncfcont1.html#T20" target="_self">Th20</a>, <a class="ref" href="ncfcont1.html#T22" target="_self">Th22</a>, , <a class="ref" href="vfunct_2.html#D2">VFUNCT_2:def 2</a></i>
<br/>.= 
 <a href="clvect_1.html#K11">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> <a href="clvect_1.html#K7">-</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ncfcont1.html#T22" target="_self">Th22</a>, , <a class="ref" href="clvect_1.html#T122">CLVECT_1:122</a></i>
<br/>.= 
 <a href="clvect_1.html#K11">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_2.html#K2">-</a> <font color="Maroon">f2</font></span>)</span> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font></span>)</span>
<b>by </b><i><a class="ref" href="ncfcont1.html#T21" target="_self">Th21</a>, <a class="ref" href="ncfcont1.html#T4" target="_self">Th4</a></i>
;<br/><b>hence </b><a NAME="E7:109_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_2.html#K2">-</a> <font color="Maroon">f2</font></span>)</span> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font> is <a href="clvect_1.html#V6">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_2.html#K2">-</a> <font color="Maroon">f2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="clvect_1.html#K11">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_2.html#K2">-</a> <font color="Maroon">f2</font></span>)</span> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="ncfcont1.html#T21" target="_self">Th21</a>, , <a class="ref" href="ncfcont1.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:109"/>
<font color="Maroon">f1</font> <a href="vfunct_2.html#K2">-</a> <font color="Maroon">f2</font> <a href="ncfcont1.html#R7" target="_self">is_continuous_on</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="ncfcont1.html#T19" target="_self">Th19</a>, <a class="ref" href="ncfcont1.html#T8" target="_self">Th8</a></i>;<br/>


</div>
