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<b>let </b><font color="Maroon" title="c1">CNS</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">RNS</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c1">CNS</font>,<font color="Maroon" title="c2">RNS</font>;<br/>







<b>assume </b><a NAME="E1:118"/><i><font color="Green" title="E92">A1</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<b>thus </b><a NAME="E2:118"/>
<span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="ncfcont1.html#R11" title="NCFCONT1:pred.11">is_continuous_on</a> <font color="Maroon" title="c3">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1"><b>proof </b></a><div class="add">

<a NAME="E1:118_1"/><i><font color="Green" title="E93">A2</font></i>: 
<font color="Maroon" title="c3">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="c4">f</font>
 
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:118"><i><font color="Green" title="E92">A1</font></i></a>, <a class="ref" href="ncfcont1.html#D22" target="_self" title="NCFCONT1:def.22">Def22</a></i>;<br/>
<b>hence </b><a NAME="E2:118_1"/><i><font color="Green" title="E94">A3</font></i>: 
<font color="Maroon" title="c3">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"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span>
 <b>by </b><i><a class="ref" href="ncfcont1.html#D3" target="_self" title="NCFCONT1:def.3">Def3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="ncfcont1.html#D25" target="_self" title="NCFCONT1:def.25">NCFCONT1:def 25</a><br/>

<b>let </b><font color="Maroon" title="c5">r</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CNS</font>;<br/>

<b>assume </b><a NAME="E3:118_1"/><i><font color="Green" title="E95">A4</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E4:118_1"/><b>then </b><i><font color="Green" title="E96">A5</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font> <a href="ncfcont1.html#R2" title="NCFCONT1:pred.2">is_continuous_in</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:118"><i><font color="Green" title="E92">A1</font></i></a>, <a class="ref" href="ncfcont1.html#D22" target="_self" title="NCFCONT1:def.22">Def22</a></i>;<br/>
<b>thus </b><a NAME="E5:118_1"/>
<span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font> <a href="ncfcont1.html#R5" title="NCFCONT1:pred.5">is_continuous_in</a> <font color="Maroon" title="c5">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:118_1_1"/><i><font color="Green" title="E97">A6</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="ncfcont1.html#E2:118_1"><i><font color="Green" title="E94">A3</font></i></a>, <a class="txt" href="ncfcont1.html#E3:118_1"><i><font color="Green" title="E95">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E2:118_1_1"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="ncfcont1.html#D19" target="_self" title="NCFCONT1:def.19">NCFCONT1:def 19</a><br/>

<b>let </b><font color="Maroon" title="c6">s1</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">CNS</font>;<br/>

<b>assume </b><a NAME="E3:118_1_1"/><i><font color="Green" title="E98">A7</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s1</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"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> &amp; <font color="Maroon" title="c6">s1</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a> &amp;  <a href="clvect_1.html#K8" title="CLVECT_1:func.8">lim</a> <font color="Maroon" title="c6">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> )
 ;<br/>

<a NAME="E4:118_1_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">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> <span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E5:118_1_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">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="c4">f</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="ref" href="ncfcont1.html#D3" target="_self" title="NCFCONT1:def.3">Def3</a></i>;<br/>
<a NAME="E6:118_1_1"/><b>then </b><i><font color="Green" title="E99">A8</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s1</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="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E7:118_1_1"/><b>then </b><i><font color="Green" title="E100">A9</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="ncfcont1.html#E4:118_1"><i><font color="Green" title="E96">A5</font></i></a>, <a class="txt" href="ncfcont1.html#E3:118_1_1"><i><font color="Green" title="E98">A7</font></i></a>, <a class="ref" href="ncfcont1.html#D16" target="_self" title="NCFCONT1:def.16">Def16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:118_1_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E2:118_1_1_1"/><b>then </b><i><font color="Green" title="E101">A10</font></i>: 
<font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E3:118_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E6:118_1_1"><i><font color="Green" title="E99">A8</font></i></a></i>;<br/><a NAME="E4:118_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/><a NAME="E5:118_1_1_1"/><b>then </b><i><font color="Green" title="E102">A11</font></i>: 
( <font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:118_1_1_1"/><b>then </b><i><font color="Green" title="E103">A12</font></i>: 
( <font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> &amp; <font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 <b>by </b><i><a class="ref" href="ncfcont1.html#D3" target="_self" title="NCFCONT1:def.3">Def3</a></i>;<br/><b>thus </b><span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">n</font> = 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</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" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#D10" title="NORMSP_1:def.10">NORMSP_1:def 10</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E6:118_1_1"><i><font color="Green" title="E99">A8</font></i></a>, <a class="ref" href="ncfcont1.html#T11" target="_self" title="NCFCONT1:th.11">Th11</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E6:118_1_1"><i><font color="Green" title="E99">A8</font></i></a>, <a class="txt" href="ncfcont1.html#E2:118_1_1_1"><i><font color="Green" title="E101">A10</font></i></a>, <a class="ref" href="partfun2.html#T32" title="PARTFUN2:th.32">PARTFUN2:32</a></i>
<br/>.= 
<span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E6:118_1_1_1"><i><font color="Green" title="E103">A12</font></i></a>, <a class="ref" href="ncfcont1.html#D3" target="_self" title="NCFCONT1:def.3">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E5:118_1_1_1"><i><font color="Green" title="E102">A11</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E3:118_1_1"><i><font color="Green" title="E98">A7</font></i></a>, <a class="txt" href="ncfcont1.html#E2:118_1_1_1"><i><font color="Green" title="E101">A10</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="ncfcont1.html#K9" title="NCFCONT1:func.9">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">n</font>
<b>by </b><i><a class="txt" href="ncfcont1.html#E3:118_1_1"><i><font color="Green" title="E98">A7</font></i></a>, <a class="ref" href="ncfcont1.html#T14" target="_self" title="NCFCONT1:th.14">Th14</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E9:118_1_1"/><b>then </b><i><font color="Green" title="E101">A13</font></i>: 
<span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="ncfcont1.html#K9" title="NCFCONT1:func.9">*</a> <font color="Maroon" title="c6">s1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>
<b>hence </b><a NAME="E10:118_1_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="ncfcont1.html#K9" title="NCFCONT1:func.9">*</a> <font color="Maroon" title="c6">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E7:118_1_1"><i><font color="Green" title="E100">A9</font></i></a>, <a class="ref" href="normsp_1.html#T39" title="NORMSP_1:th.39">NORMSP_1:39</a></i>;<br/>

<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">r</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> = 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">r</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:118_1"><i><font color="Green" title="E93">A2</font></i></a>, <a class="txt" href="ncfcont1.html#E3:118_1"><i><font color="Green" title="E95">A4</font></i></a>, <a class="ref" href="partfun2.html#T35" title="PARTFUN2:th.35">PARTFUN2:35</a></i>
<br/>.= 
<span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">r</font>
<b>by </b><i><a class="txt" href="ncfcont1.html#E2:118_1"><i><font color="Green" title="E94">A3</font></i></a>, <a class="txt" href="ncfcont1.html#E3:118_1"><i><font color="Green" title="E95">A4</font></i></a>, <a class="ref" href="ncfcont1.html#D3" target="_self" title="NCFCONT1:def.3">Def3</a></i>
<br/>.= 
<span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">r</font>
<b>by </b><i><a class="txt" href="ncfcont1.html#E2:118_1"><i><font color="Green" title="E94">A3</font></i></a>, <a class="txt" href="ncfcont1.html#E3:118_1"><i><font color="Green" title="E95">A4</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">r</font>
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:118_1_1"><i><font color="Green" title="E97">A6</font></i></a>, <a class="ref" href="partfun2.html#T34" title="PARTFUN2:th.34">PARTFUN2:34</a></i>
;<br/>
<b>hence </b><a NAME="E12:118_1_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">r</font> <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"><span class="p3"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="ncfcont1.html#K9" title="NCFCONT1:func.9">*</a> <font color="Maroon" title="c6">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E7:118_1_1"><i><font color="Green" title="E100">A9</font></i></a>, <a class="txt" href="ncfcont1.html#E9:118_1_1"><i><font color="Green" title="E101">A13</font></i></a>, <a class="ref" href="lopban_1.html#T24" title="LOPBAN_1:th.24">LOPBAN_1:24</a></i>;<br/>


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


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

<a NAME="E3:118"/>
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c4">f</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:118"><i><font color="Green" title="E92">A1</font></i></a>, <a class="ref" href="ncfcont1.html#T90" target="_self" title="NCFCONT1:th.90">Th90</a></i>;<br/>
<b>hence </b><a NAME="E4:118"/>
 <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">-</a> <font color="Maroon" title="c4">f</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="ref" href="vfunct_1.html#T29" title="VFUNCT_1:th.29">VFUNCT_1:29</a></i>;<br/>


</div>
