<?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">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>let </b><font color="Maroon">x0</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS1</font>;<br/>







<b>assume </b><a NAME="E1:73"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">f1</font> <a href="ncfcont1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">f2</font> <a href="ncfcont1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">x0</font> )
 ;<br/>

<a NAME="E2:73"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp; ( for <font color="Olive">s1</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">CNS1</font>  st  <a href="relset_1.html#K5">rng</a> <font color="Olive">s1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp; <font color="Olive">s1</font> is <a href="clvect_1.html#V6">convergent</a> &amp;  <a href="clvect_1.html#K11">lim</a> <font color="Olive">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> holds <br/>( <font color="Maroon">f1</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Olive">s1</font> is <a href="clvect_1.html#V6">convergent</a> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font> <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="Olive">s1</font></span>)</span> ) ) )
 
<b>by </b><i/>;<br/>
<a NAME="E3:73"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> &amp; ( for <font color="Olive">s1</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">CNS1</font>  st  <a href="relset_1.html#K5">rng</a> <font color="Olive">s1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> &amp; <font color="Olive">s1</font> is <a href="clvect_1.html#V6">convergent</a> &amp;  <a href="clvect_1.html#K11">lim</a> <font color="Olive">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> holds <br/>( <font color="Maroon">f2</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Olive">s1</font> is <a href="clvect_1.html#V6">convergent</a> &amp; <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font> <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="Olive">s1</font></span>)</span> ) ) )
 
<b>by </b><i><a class="ref" href="ncfcont1.html#D16" target="_self">Def16</a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>now </b></a><div class="add"><a NAME="E1:73_1"/>
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</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#D17" target="_self">Def17</a>, <a class="ref" href="ncfcont1.html#D18" target="_self">Def18</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E2:73_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</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/><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="E3:73_1"/><i><font color="Green">E109</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</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> &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#R1">=</a> <font color="Maroon">x0</font> )
 ;<br/><a NAME="E4:73_1"/><b>then </b><i><font color="Green">E110</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="vfunct_2.html#D1">VFUNCT_2:def 1</a></i>;<br/><a NAME="E5:73_1"/>
 <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> <a href="hidden.html#R1">=</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="vfunct_2.html#D1">VFUNCT_2:def 1</a></i>;<br/><a NAME="E6:73_1"/><b>then </b>
 <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> <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="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E7:73_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</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#D20" target="_self">Def20</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:73_1"/><b>then </b><i><font color="Green">E112</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> <font color="Maroon">x0</font> <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#D16" target="_self">Def16</a>, <a class="ref" href="ncfcont1.html#D20" target="_self">Def20</a>, </i>;<br/><a NAME="E9:73_1"/>
 <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> <a href="hidden.html#R1">=</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="vfunct_2.html#D1">VFUNCT_2:def 1</a></i>;<br/><a NAME="E10:73_1"/><b>then </b>
 <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> <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="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E11:73_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</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#D20" target="_self">Def20</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E12:73_1"/><b>then </b><i><font color="Green">E113</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> <font color="Maroon">x0</font> <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#D16" target="_self">Def16</a>, <a class="ref" href="ncfcont1.html#D20" target="_self">Def20</a>, </i>;<br/><a NAME="E13:73_1"/><b>then </b>
<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="clvect_1.html#T115">CLVECT_1:115</a></i>;<br/><b>hence </b><a NAME="E14:73_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>
 <b>by </b><i><a class="ref" href="ncfcont1.html#T5" target="_self">Th5</a>, <a class="ref" href="ncfcont1.html#T4" target="_self">Th4</a></i>;<br/><b>thus </b><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> <font color="Maroon">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i><a class="ref" href="ncfcont1.html#D19" target="_self">Def19</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#T6" target="_self">Th6</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#T5" target="_self">Th5</a>, <a class="ref" href="ncfcont1.html#T4" target="_self">Th4</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:73"/>
<font color="Maroon">f1</font> <a href="vfunct_2.html#K1">+</a> <font color="Maroon">f2</font> <a href="ncfcont1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">x0</font>
 <b>by </b><i/>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2"><b>now </b></a><div class="add"><a NAME="E1:73_2"/>
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</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#D17" target="_self">Def17</a>, <a class="ref" href="ncfcont1.html#D18" target="_self">Def18</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E2:73_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</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="vfunct_2.html#D2">VFUNCT_2:def 2</a></i>;<br/><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="E3:73_2"/><i><font color="Green">E115</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</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> &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#R1">=</a> <font color="Maroon">x0</font> )
 ;<br/><a NAME="E4:73_2"/><b>then </b><i><font color="Green">E116</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="vfunct_2.html#D2">VFUNCT_2:def 2</a></i>;<br/><a NAME="E5:73_2"/>
 <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> <a href="hidden.html#R1">=</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="vfunct_2.html#D2">VFUNCT_2:def 2</a></i>;<br/><a NAME="E6:73_2"/><b>then </b>
 <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> <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="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E7:73_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</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#D20" target="_self">Def20</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:73_2"/><b>then </b><i><font color="Green">E118</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> <font color="Maroon">x0</font> <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#D16" target="_self">Def16</a>, <a class="ref" href="ncfcont1.html#D20" target="_self">Def20</a>, </i>;<br/><a NAME="E9:73_2"/>
 <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> <a href="hidden.html#R1">=</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="vfunct_2.html#D2">VFUNCT_2:def 2</a></i>;<br/><a NAME="E10:73_2"/><b>then </b>
 <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> <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="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E11:73_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</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#D20" target="_self">Def20</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E12:73_2"/><b>then </b><i><font color="Green">E119</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> <font color="Maroon">x0</font> <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#D16" target="_self">Def16</a>, <a class="ref" href="ncfcont1.html#D20" target="_self">Def20</a>, </i>;<br/><a NAME="E13:73_2"/><b>then </b>
<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="clvect_1.html#T116">CLVECT_1:116</a></i>;<br/><b>hence </b><a NAME="E14:73_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>
 <b>by </b><i><a class="ref" href="ncfcont1.html#T5" target="_self">Th5</a>, <a class="ref" href="ncfcont1.html#T4" target="_self">Th4</a></i>;<br/><b>thus </b><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> <font color="Maroon">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i><a class="ref" href="ncfcont1.html#D19" target="_self">Def19</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#T6" target="_self">Th6</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#T5" target="_self">Th5</a>, <a class="ref" href="ncfcont1.html#T4" target="_self">Th4</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:73"/>
<font color="Maroon">f1</font> <a href="vfunct_2.html#K2">-</a> <font color="Maroon">f2</font> <a href="ncfcont1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">x0</font>
 <b>by </b><i/>;<br/>


</div>
