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

<b>let </b><font color="Maroon" title="c1">CNS1</font>, <font color="Maroon" title="c2">CNS2</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">f2</font> be   <a href="nfcont_1.html#NM1" title="NFCONT_1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">CNS1</font>,<font color="Maroon" title="c2">CNS2</font>;<br/><b>let </b><font color="Maroon" title="c5">x0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CNS1</font>;<br/>







<b>assume </b><a NAME="E1:73"/><i><font color="Green" title="E66">A1</font></i>: 
( <font color="Maroon" title="c3">f1</font> <a href="ncfcont1.html#R1" title="NCFCONT1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x0</font> &amp; <font color="Maroon" title="c4">f2</font> <a href="ncfcont1.html#R1" title="NCFCONT1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x0</font> )
 ;<br/>

<a NAME="E2:73"/><b>then </b><i><font color="Green" title="E67">A2</font></i>: 
( <font color="Maroon" title="c5">x0</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="c3">f1</font> &amp; ( for <font color="Olive" title="b1">s1</font> being  <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">CNS1</font>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">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> <font color="Maroon" title="c3">f1</font> &amp; <font color="Olive" title="b1">s1</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a> &amp;  <a href="clvect_1.html#K11" title="CLVECT_1:func.11">lim</a> <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x0</font> holds <br/>( <font color="Maroon" title="c3">f1</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Olive" title="b1">s1</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a> &amp; <font color="Maroon" title="c3">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clvect_1.html#K11" title="CLVECT_1:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Olive" title="b1">s1</font></span>)</span> ) ) )
 
<b>by </b><i><a class="ref" href="ncfcont1.html#D15" target="_self" title="NCFCONT1:def.15">Def15</a></i>;<br/>
<a NAME="E3:73"/><i><font color="Green" title="E68">A3</font></i>: 
( <font color="Maroon" title="c5">x0</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">f2</font> &amp; ( for <font color="Olive" title="b1">s1</font> being  <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">CNS1</font>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">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> <font color="Maroon" title="c4">f2</font> &amp; <font color="Olive" title="b1">s1</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a> &amp;  <a href="clvect_1.html#K11" title="CLVECT_1:func.11">lim</a> <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x0</font> holds <br/>( <font color="Maroon" title="c4">f2</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Olive" title="b1">s1</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a> &amp; <font color="Maroon" title="c4">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clvect_1.html#K11" title="CLVECT_1:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Olive" title="b1">s1</font></span>)</span> ) ) )
 
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:73"><i><font color="Green" title="E66">A1</font></i></a>, <a class="ref" href="ncfcont1.html#D15" target="_self" title="NCFCONT1:def.15">Def15</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" title="c5">x0</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="c3">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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">f2</font></span>)</span>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E2:73"><i><font color="Green" title="E67">A2</font></i></a>, <a class="txt" href="ncfcont1.html#E3:73"><i><font color="Green" title="E68">A3</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="E2:73_1"/><i><font color="Green" title="E69">A4</font></i>: 
<font color="Maroon" title="c5">x0</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="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_2.html#D1" title="VFUNCT_2:def.1">VFUNCT_2:def 1</a></i>;<br/><b>let </b><font color="Maroon" title="c6">s1</font> be   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">CNS1</font>;<br/><b>assume </b><a NAME="E3:73_1"/><i><font color="Green" title="E70">A5</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="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c4">f2</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#K11" title="CLVECT_1:func.11">lim</a> <font color="Maroon" title="c6">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x0</font> )
 ;<br/><a NAME="E4:73_1"/><b>then </b><i><font color="Green" title="E71">A6</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> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_2.html#D1" title="VFUNCT_2:def.1">VFUNCT_2:def 1</a></i>;<br/><a NAME="E5:73_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_2.html#D1" title="VFUNCT_2:def.1">VFUNCT_2:def 1</a></i>;<br/><a NAME="E6:73_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <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="c3">f1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E7:73_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>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f1</font>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E3:73_1"><i><font color="Green" title="E70">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:73_1"/><b>then </b><i><font color="Green" title="E72">A7</font></i>: 
( <font color="Maroon" title="c3">f1</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a> &amp; <font color="Maroon" title="c3">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clvect_1.html#K11" title="CLVECT_1:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <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#E1:73"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="ncfcont1.html#E3:73_1"><i><font color="Green" title="E70">A5</font></i></a>, <a class="ref" href="ncfcont1.html#D15" target="_self" title="NCFCONT1:def.15">Def15</a></i>;<br/><a NAME="E9:73_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_2.html#D1" title="VFUNCT_2:def.1">VFUNCT_2:def 1</a></i>;<br/><a NAME="E10:73_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <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">f2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E11:73_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>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f2</font>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E3:73_1"><i><font color="Green" title="E70">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E12:73_1"/><b>then </b><i><font color="Green" title="E73">A8</font></i>: 
( <font color="Maroon" title="c4">f2</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a> &amp; <font color="Maroon" title="c4">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clvect_1.html#K11" title="CLVECT_1:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <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#E1:73"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="ncfcont1.html#E3:73_1"><i><font color="Green" title="E70">A5</font></i></a>, <a class="ref" href="ncfcont1.html#D15" target="_self" title="NCFCONT1:def.15">Def15</a></i>;<br/><a NAME="E13:73_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="clvect_1.html#K6" title="CLVECT_1:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E8:73_1"><i><font color="Green" title="E72">A7</font></i></a>, <a class="ref" href="clvect_1.html#T115" title="CLVECT_1:th.115">CLVECT_1:115</a></i>;<br/><b>hence </b><a NAME="E14:73_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E4:73_1"><i><font color="Green" title="E71">A6</font></i></a>, <a class="ref" href="ncfcont1.html#T44" target="_self" title="NCFCONT1:th.44">Th44</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x0</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E2:73_1"><i><font color="Green" title="E69">A4</font></i></a>, <a class="ref" href="vfunct_2.html#D1" title="VFUNCT_2:def.1">VFUNCT_2:def 1</a></i>
<br/>.= 
 <a href="clvect_1.html#K11" title="CLVECT_1:func.11">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="clvect_1.html#K6" title="CLVECT_1:func.6">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E8:73_1"><i><font color="Green" title="E72">A7</font></i></a>, <a class="txt" href="ncfcont1.html#E12:73_1"><i><font color="Green" title="E73">A8</font></i></a>, <a class="ref" href="clvect_1.html#T121" title="CLVECT_1:th.121">CLVECT_1:121</a></i>
<br/>.= 
 <a href="clvect_1.html#K11" title="CLVECT_1:func.11">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c4">f2</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#E4:73_1"><i><font color="Green" title="E71">A6</font></i></a>, <a class="ref" href="ncfcont1.html#T44" target="_self" title="NCFCONT1:th.44">Th44</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:73"/>
<font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c4">f2</font> <a href="ncfcont1.html#R1" title="NCFCONT1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x0</font>
 <b>by </b><i><a class="ref" href="ncfcont1.html#D15" target="_self" title="NCFCONT1:def.15">Def15</a></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" title="c5">x0</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="c3">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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">f2</font></span>)</span>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E2:73"><i><font color="Green" title="E67">A2</font></i></a>, <a class="txt" href="ncfcont1.html#E3:73"><i><font color="Green" title="E68">A3</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="E2:73_2"/><i><font color="Green" title="E69">A9</font></i>: 
<font color="Maroon" title="c5">x0</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="c3">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_2.html#D2" title="VFUNCT_2:def.2">VFUNCT_2:def 2</a></i>;<br/><b>let </b><font color="Maroon" title="c6">s1</font> be   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">CNS1</font>;<br/><b>assume </b><a NAME="E3:73_2"/><i><font color="Green" title="E70">A10</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="c3">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c4">f2</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#K11" title="CLVECT_1:func.11">lim</a> <font color="Maroon" title="c6">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x0</font> )
 ;<br/><a NAME="E4:73_2"/><b>then </b><i><font color="Green" title="E71">A11</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> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_2.html#D2" title="VFUNCT_2:def.2">VFUNCT_2:def 2</a></i>;<br/><a NAME="E5:73_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_2.html#D2" title="VFUNCT_2:def.2">VFUNCT_2:def 2</a></i>;<br/><a NAME="E6:73_2"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <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="c3">f1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E7:73_2"/><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>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f1</font>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E3:73_2"><i><font color="Green" title="E70">A10</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:73_2"/><b>then </b><i><font color="Green" title="E72">A12</font></i>: 
( <font color="Maroon" title="c3">f1</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a> &amp; <font color="Maroon" title="c3">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clvect_1.html#K11" title="CLVECT_1:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <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#E1:73"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="ncfcont1.html#E3:73_2"><i><font color="Green" title="E70">A10</font></i></a>, <a class="ref" href="ncfcont1.html#D15" target="_self" title="NCFCONT1:def.15">Def15</a></i>;<br/><a NAME="E9:73_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_2.html#D2" title="VFUNCT_2:def.2">VFUNCT_2:def 2</a></i>;<br/><a NAME="E10:73_2"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <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">f2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E11:73_2"/><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>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f2</font>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E3:73_2"><i><font color="Green" title="E70">A10</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E12:73_2"/><b>then </b><i><font color="Green" title="E73">A13</font></i>: 
( <font color="Maroon" title="c4">f2</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a> &amp; <font color="Maroon" title="c4">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clvect_1.html#K11" title="CLVECT_1:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <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#E1:73"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="ncfcont1.html#E3:73_2"><i><font color="Green" title="E70">A10</font></i></a>, <a class="ref" href="ncfcont1.html#D15" target="_self" title="NCFCONT1:def.15">Def15</a></i>;<br/><a NAME="E13:73_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E8:73_2"><i><font color="Green" title="E72">A12</font></i></a>, <a class="ref" href="clvect_1.html#T116" title="CLVECT_1:th.116">CLVECT_1:116</a></i>;<br/><b>hence </b><a NAME="E14:73_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E4:73_2"><i><font color="Green" title="E71">A11</font></i></a>, <a class="ref" href="ncfcont1.html#T44" target="_self" title="NCFCONT1:th.44">Th44</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x0</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E2:73_2"><i><font color="Green" title="E69">A9</font></i></a>, <a class="ref" href="vfunct_2.html#D2" title="VFUNCT_2:def.2">VFUNCT_2:def 2</a></i>
<br/>.= 
 <a href="clvect_1.html#K11" title="CLVECT_1:func.11">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E8:73_2"><i><font color="Green" title="E72">A12</font></i></a>, <a class="txt" href="ncfcont1.html#E12:73_2"><i><font color="Green" title="E73">A13</font></i></a>, <a class="ref" href="clvect_1.html#T122" title="CLVECT_1:th.122">CLVECT_1:122</a></i>
<br/>.= 
 <a href="clvect_1.html#K11" title="CLVECT_1:func.11">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c4">f2</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#E4:73_2"><i><font color="Green" title="E71">A11</font></i></a>, <a class="ref" href="ncfcont1.html#T44" target="_self" title="NCFCONT1:th.44">Th44</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:73"/>
<font color="Maroon" title="c3">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c4">f2</font> <a href="ncfcont1.html#R1" title="NCFCONT1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x0</font>
 <b>by </b><i><a class="ref" href="ncfcont1.html#D15" target="_self" title="NCFCONT1:def.15">Def15</a></i>;<br/>


</div>
