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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">T</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>







<b>assume </b><a NAME="E1:48"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> <a href="nfcont_1.html#R3" target="_self">is_continuous_on</a> <font color="Maroon">X</font>
 ;<br/>

<b>thus </b><a NAME="E2:48"/>
<span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="nfcont_1.html#R4" target="_self">is_continuous_on</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<a NAME="E1:48_1"/><i><font color="Green">E44</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">f</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E2:48_1"/><i><font color="Green">E55</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"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="nfcont_1.html#D12" target="_self">NFCONT_1:def 12</a><br/>

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

<b>assume </b><a NAME="E3:48_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E4:48_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font> <a href="nfcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">r</font>
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E5:48_1"/>
<span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font> <a href="nfcont_1.html#R2" target="_self">is_continuous_in</a> <font color="Maroon">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:48_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="nfcont_1.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E2:48_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nfcont_1.html#D10" target="_self">NFCONT_1:def 10</a><br/>

<b>let </b><font color="Maroon">s1</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E3:48_1_1"/><i><font color="Green">E81</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"><span class="p2"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">s1</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 ;<br/>

<a NAME="E4:48_1_1"/><b>then </b>
 <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> <span class="p2"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E5:48_1_1"/><b>then </b>
 <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">f</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:48_1_1"/><b>then </b><i><font color="Green">E82</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">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E7:48_1_1"/><b>then </b><i><font color="Green">E83</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">s1</font> is <a href="normsp_1.html#V4">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> )
 
<b>by </b><i>, <a class="txt" href="nfcont_1.html#E1:48_1"><i><font color="Green">E44</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:48_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">s1</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="E2:48_1_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E3:48_1_1_1"/><b>then </b>
<font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</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">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:48_1_1_1"/><b>then </b>
<font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</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">f</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E5:48_1_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
( <font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:48_1_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
( <font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> &amp; <font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/><b>thus </b><span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">s1</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#D10">NORMSP_1:def 10</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i>, <a class="ref" href="nfcont_1.html#T2" target="_self">Th2</a>, <a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>
<br/>.= 
<span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="nfcont_1.html#D2" target="_self">Def2</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E1:48_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="nfcont_1.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="nfcont_1.html#K4" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="nfcont_1.html#E1:48_1"><i><font color="Green">E44</font></i></a>, </i>
;<br/></div><b>end;</b></div>
<a NAME="E9:48_1_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">s1</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="nfcont_1.html#K4" target="_self">*</a> <font color="Maroon">s1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<b>hence </b><a NAME="E10:48_1_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="nfcont_1.html#K4" target="_self">*</a> <font color="Maroon">s1</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i>, <a class="ref" href="normsp_1.html#T39">NORMSP_1:39</a></i>;<br/>

<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">r</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> = 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">r</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i>, <a class="ref" href="nfcont_1.html#D1" target="_self">Def1</a>, <a class="ref" href="partfun2.html#T35">PARTFUN2:35</a></i>
<br/>.= 
<span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font>
<b>by </b><i>, <a class="ref" href="nfcont_1.html#D1" target="_self">Def1</a>, </i>
<br/>.= 
<span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">r</font>
<b>by </b><i>, <a class="ref" href="nfcont_1.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">r</font>
<b>by </b><i><a class="txt" href="nfcont_1.html#E1:48"><i><font color="Green">E43</font></i></a>, <a class="ref" href="partfun2.html#T34">PARTFUN2:34</a></i>
;<br/>
<b>hence </b><a NAME="E12:48_1_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="nfcont_1.html#K4" target="_self">*</a> <font color="Maroon">s1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nfcont_1.html#T2" target="_self">Th2</a>, <a class="ref" href="lopban_1.html#T24">LOPBAN_1:24</a></i>;<br/>


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


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

<a NAME="E3:48"/>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="vfunct_1.html#K4">(#)</a> <font color="Maroon">f</font> <a href="nfcont_1.html#R3" target="_self">is_continuous_on</a> <font color="Maroon">X</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:48"/>
 <a href="vfunct_1.html#K6">-</a> <font color="Maroon">f</font> <a href="nfcont_1.html#R3" target="_self">is_continuous_on</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="vfunct_1.html#T29">VFUNCT_1:29</a></i>;<br/>


</div>
