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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>







<b>thus </b><a NAME="E1:88"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="ncfcont1.html#R7" target="_self">is_continuous_on</a> <font color="Maroon">c<sub>3</sub></font> implies ( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> st  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="clvect_1.html#V6">convergent</a> &amp;  <a href="clvect_1.html#K11">lim</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Olive">b<sub>1</sub></font> is <a href="clvect_1.html#V6">convergent</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:88_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="ncfcont1.html#R7" target="_self">is_continuous_on</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:88_1"/><b>then </b><i><font color="Green">E77</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font> <a href="ncfcont1.html#R1" target="_self">is_continuous_in</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="ref" href="ncfcont1.html#D21" target="_self">Def21</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:88_1_1"/><i><font color="Green">E78</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="clvect_1.html#V6">convergent</a> &amp;  <a href="clvect_1.html#K11">lim</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><i><font color="Green">E79</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="ncfcont1.html#E2:88_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><a NAME="E3:88_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font> <a href="ncfcont1.html#R1" target="_self">is_continuous_in</a>  <a href="clvect_1.html#K11">lim</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E1:88_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="ncfcont1.html#E1:88_1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="ncfcont1.html#D21" target="_self">Def21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:88_1_1_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E2:88_1_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E3:88_1_1_2"/><i><font color="Green">E82</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E2:88_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="ncfcont1.html#E1:88_1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:88_1_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="ncfcont1.html#E2:88_1_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="ncfcont1.html#T10" target="_self">Th10</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:88_1_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="ncfcont1.html#E2:88_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="ncfcont1.html#E2:88_1_1_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="ncfcont1.html#E3:88_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="ncfcont1.html#T10" target="_self">Th10</a></i>
;<br/></div><b>end;</b></div><a NAME="E5:88_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></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">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:88_1_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="ncfcont1.html#E2:88_1_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>
<br/>.= 
 <a href="clvect_1.html#K11">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:88_1_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="ncfcont1.html#E2:88_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="ncfcont1.html#E3:88_1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="ncfcont1.html#E5:88_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="ncfcont1.html#D15" target="_self">Def15</a></i>
;<br/><b>hence </b><a NAME="E7:88_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> is <a href="clvect_1.html#V6">convergent</a> &amp; <font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></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">c<sub>4</sub></font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="ncfcont1.html#E1:88_1_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="ncfcont1.html#E2:88_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="ncfcont1.html#E3:88_1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="ncfcont1.html#E5:88_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="ncfcont1.html#D15" target="_self">Def15</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:88_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> st  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="clvect_1.html#V6">convergent</a> &amp;  <a href="clvect_1.html#K11">lim</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Olive">b<sub>1</sub></font> is <a href="clvect_1.html#V6">convergent</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="ncfcont1.html#E1:88_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="ncfcont1.html#D21" target="_self">Def21</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:88"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> st  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="clvect_1.html#V6">convergent</a> &amp;  <a href="clvect_1.html#K11">lim</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Olive">b<sub>1</sub></font> is <a href="clvect_1.html#V6">convergent</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:88_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><i><font color="Green">E78</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="ncfcont1.html#E2:88"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:88_2_2"/><i><font color="Green">E79</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="clvect_1.html#V6">convergent</a> &amp;  <a href="clvect_1.html#K11">lim</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:88_2_2_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E2:88_2_2_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E3:88_2_2_1"/><i><font color="Green">E81</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E2:88"><i><font color="Green">E76</font></i></a>, <a class="txt" href="ncfcont1.html#E2:88_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="ncfcont1.html#E1:88_2_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:88_2_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="ncfcont1.html#T10" target="_self">Th10</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:88_2_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="ncfcont1.html#E2:88_2_2_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="ncfcont1.html#E3:88_2_2_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="ncfcont1.html#T10" target="_self">Th10</a></i>
;<br/></div><b>end;</b></div><a NAME="E3:88_2_2"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></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">c<sub>6</sub></font></span>)</span> = 
<font color="Maroon">c<sub>4</sub></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">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:88_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="ncfcont1.html#E2:88_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="ncfcont1.html#E1:88_2_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="partfun2.html#T32">PARTFUN2:32</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">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E2:88"><i><font color="Green">E76</font></i></a>, <a class="txt" href="ncfcont1.html#E1:88_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="ncfcont1.html#E2:88_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="ncfcont1.html#E1:88_2_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="ncfcont1.html#E3:88_2_2"><i><font color="Green">E80</font></i></a></i>
;<br/><b>hence </b><a NAME="E5:88_2_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> is <a href="clvect_1.html#V6">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font> <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">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="ncfcont1.html#E2:88"><i><font color="Green">E76</font></i></a>, <a class="txt" href="ncfcont1.html#E1:88_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="ncfcont1.html#E2:88_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="ncfcont1.html#E1:88_2_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="ncfcont1.html#E3:88_2_2"><i><font color="Green">E80</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:88_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font> <a href="ncfcont1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E1:88_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="ncfcont1.html#E2:88_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="ncfcont1.html#D15" target="_self">Def15</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:88"/>
<font color="Maroon">c<sub>4</sub></font> <a href="ncfcont1.html#R7" target="_self">is_continuous_on</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E2:88"><i><font color="Green">E76</font></i></a>, <a class="ref" href="ncfcont1.html#D21" target="_self">Def21</a></i>;<br/>


</div>
