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

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font> being  <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c1">CNS1</font>,<font color="Maroon" title="c2">CNS2</font>  st <font color="Olive" title="b2">f1</font> <a href="ncfcont1.html#R7" title="NCFCONT1:pred.7">is_continuous_on</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b3">f2</font> <a href="ncfcont1.html#R7" title="NCFCONT1:pred.7">is_continuous_on</a> <font color="Olive" title="b1">X</font> holds <br/>( <font color="Olive" title="b2">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Olive" title="b3">f2</font> <a href="ncfcont1.html#R7" title="NCFCONT1:pred.7">is_continuous_on</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b2">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Olive" title="b3">f2</font> <a href="ncfcont1.html#R7" title="NCFCONT1:pred.7">is_continuous_on</a> <font color="Olive" title="b1">X</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being  <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c1">CNS1</font>,<font color="Maroon" title="c2">CNS2</font>  st <font color="Olive" title="b1">f1</font> <a href="ncfcont1.html#R7" title="NCFCONT1:pred.7">is_continuous_on</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b2">f2</font> <a href="ncfcont1.html#R7" title="NCFCONT1:pred.7">is_continuous_on</a> <font color="Maroon" title="c3">X</font> holds <br/>( <font color="Olive" title="b1">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Olive" title="b2">f2</font> <a href="ncfcont1.html#R7" title="NCFCONT1:pred.7">is_continuous_on</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b1">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Olive" title="b2">f2</font> <a href="ncfcont1.html#R7" title="NCFCONT1:pred.7">is_continuous_on</a> <font color="Maroon" title="c3">X</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">f2</font> be   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c1">CNS1</font>,<font color="Maroon" title="c2">CNS2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">f1</font> <a href="ncfcont1.html#R7" title="NCFCONT1:pred.7">is_continuous_on</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="ncfcont1.html#R7" title="NCFCONT1:pred.7">is_continuous_on</a> <font color="Maroon" title="c3">X</font> implies ( <font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f2</font> <a href="ncfcont1.html#R7" title="NCFCONT1:pred.7">is_continuous_on</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c5">f2</font> <a href="ncfcont1.html#R7" title="NCFCONT1:pred.7">is_continuous_on</a> <font color="Maroon" title="c3">X</font> ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:101"/><span class="lab"><font color="Green" title="E60">A1</font></span>: 
( <font color="Maroon" title="c4">f1</font> <a href="ncfcont1.html#R7" title="NCFCONT1:pred.7">is_continuous_on</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="ncfcont1.html#R7" title="NCFCONT1:pred.7">is_continuous_on</a> <font color="Maroon" title="c3">X</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f2</font> <a href="ncfcont1.html#R7" title="NCFCONT1:pred.7">is_continuous_on</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c5">f2</font> <a href="ncfcont1.html#R7" title="NCFCONT1:pred.7">is_continuous_on</a> <font color="Maroon" title="c3">X</font> )</span><br/>

<a NAME="E2:101"/><span class="kw">then </span>
( <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f1</font> &amp; <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ncfcont1.html#T62" target="_self" title="NCFCONT1:th.62">Th62</a></span>;<br/>
<a NAME="E3:101"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A2</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/>
<a NAME="E4:101"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A3</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vfunct_2.html#D1" title="VFUNCT_2:def.1">VFUNCT_2:def 1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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">CNS1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c6">s1</font> is  <a href="clvect_1.html#V9" title="CLVECT_1:attr.9">convergent</a>  &amp;  <a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <font color="Maroon" title="c6">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font> is  <a href="clvect_1.html#V9" title="CLVECT_1:attr.9">convergent</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:101_1"/><span class="lab"><font color="Green" title="E63">A4</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">and </span><br/><a NAME="E2:101_1"/><span class="lab"><font color="Green" title="E64">A5</font></span>: 
<font color="Maroon" title="c6">s1</font> is  <a href="clvect_1.html#V9" title="CLVECT_1:attr.9">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E3:101_1"/><span class="lab"><font color="Green" title="E65">A6</font></span>: 
 <a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <font color="Maroon" title="c6">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font> is  <a href="clvect_1.html#V9" title="CLVECT_1:attr.9">convergent</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span> )</span><br/><a NAME="E4:101_1"/><span class="lab"><font color="Green" title="E66">A7</font></span>: 
( <font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font> is  <a href="clvect_1.html#V9" title="CLVECT_1:attr.9">convergent</a>  &amp; <font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font> is  <a href="clvect_1.html#V9" title="CLVECT_1:attr.9">convergent</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E1:101_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E2:101_1"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="txt" href="#E3:101_1"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="ref" href="ncfcont1.html#T62" target="_self" title="NCFCONT1:th.62">Th62</a></span>;<br/><a NAME="E5:101_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span> is  <a href="clvect_1.html#V9" title="CLVECT_1:attr.9">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#T115" title="CLVECT_1:th.115">CLVECT_1:115</a></span>;<br/><a NAME="E6:101_1"/><span class="lab"><font color="Green" title="E68">A9</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="txt" href="#E1:101_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E7:101_1"/>
( <font color="Maroon" title="c4">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span> &amp; <font color="Maroon" title="c5">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E1:101_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E2:101_1"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="txt" href="#E3:101_1"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="ref" href="ncfcont1.html#T62" target="_self" title="NCFCONT1:th.62">Th62</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <font color="Maroon" title="c6">s1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="txt" href="#E3:101_1"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="ref" href="vfunct_2.html#D1" title="VFUNCT_2:def.1">VFUNCT_2:def 1</a></span>
<br/>.= 
 <a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="ref" href="clvect_1.html#T121" title="CLVECT_1:th.121">CLVECT_1:121</a></span>
<br/>.= 
 <a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:101_1"><span class="lab"><font color="Green" title="E68">A9</font></span></a>, <a class="ref" href="ncfcont1.html#T44" target="_self" title="NCFCONT1:th.44">Th44</a></span>
;<br/><span class="kw">hence </span><a NAME="E9:101_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font> is  <a href="clvect_1.html#V9" title="CLVECT_1:attr.9">convergent</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:101_1"><span class="lab"><font color="Green" title="E68">A9</font></span></a>, <a class="txt" href="#E5:101_1"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="ref" href="ncfcont1.html#T44" target="_self" title="NCFCONT1:th.44">Th44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:101"/>
<font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">+</a> <font color="Maroon" title="c5">f2</font> <a href="ncfcont1.html#R7" title="NCFCONT1:pred.7">is_continuous_on</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="ref" href="ncfcont1.html#T62" target="_self" title="NCFCONT1:th.62">Th62</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c5">f2</font> <a href="ncfcont1.html#R7" title="NCFCONT1:pred.7">is_continuous_on</a> <font color="Maroon" title="c3">X</font></span><br/>

<a NAME="E7:101"/><span class="lab"><font color="Green" title="E63">A10</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="ref" href="vfunct_2.html#D2" title="VFUNCT_2:def.2">VFUNCT_2:def 2</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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">CNS1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c6">s1</font> is  <a href="clvect_1.html#V9" title="CLVECT_1:attr.9">convergent</a>  &amp;  <a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <font color="Maroon" title="c6">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font> is  <a href="clvect_1.html#V9" title="CLVECT_1:attr.9">convergent</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:101_2"/><span class="lab"><font color="Green" title="E64">A11</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">and </span><br/><a NAME="E2:101_2"/><span class="lab"><font color="Green" title="E65">A12</font></span>: 
<font color="Maroon" title="c6">s1</font> is  <a href="clvect_1.html#V9" title="CLVECT_1:attr.9">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E3:101_2"/><span class="lab"><font color="Green" title="E66">A13</font></span>: 
 <a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <font color="Maroon" title="c6">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font> is  <a href="clvect_1.html#V9" title="CLVECT_1:attr.9">convergent</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span> )</span><br/><a NAME="E4:101_2"/><span class="lab"><font color="Green" title="E67">A14</font></span>: 
( <font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font> is  <a href="clvect_1.html#V9" title="CLVECT_1:attr.9">convergent</a>  &amp; <font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font> is  <a href="clvect_1.html#V9" title="CLVECT_1:attr.9">convergent</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E1:101_2"><span class="lab"><font color="Green" title="E64">A11</font></span></a>, <a class="txt" href="#E2:101_2"><span class="lab"><font color="Green" title="E65">A12</font></span></a>, <a class="txt" href="#E3:101_2"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="ref" href="ncfcont1.html#T62" target="_self" title="NCFCONT1:th.62">Th62</a></span>;<br/><a NAME="E5:101_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A15</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span> is  <a href="clvect_1.html#V9" title="CLVECT_1:attr.9">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#T116" title="CLVECT_1:th.116">CLVECT_1:116</a></span>;<br/><a NAME="E6:101_2"/><span class="lab"><font color="Green" title="E69">A16</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="txt" href="#E1:101_2"><span class="lab"><font color="Green" title="E64">A11</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E7:101_2"/>
( <font color="Maroon" title="c4">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span> &amp; <font color="Maroon" title="c5">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E1:101_2"><span class="lab"><font color="Green" title="E64">A11</font></span></a>, <a class="txt" href="#E2:101_2"><span class="lab"><font color="Green" title="E65">A12</font></span></a>, <a class="txt" href="#E3:101_2"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="ref" href="ncfcont1.html#T62" target="_self" title="NCFCONT1:th.62">Th62</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <font color="Maroon" title="c6">s1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101"><span class="lab"><font color="Green" title="E63">A10</font></span></a>, <a class="txt" href="#E3:101_2"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="ref" href="vfunct_2.html#D2" title="VFUNCT_2:def.2">VFUNCT_2:def 2</a></span>
<br/>.= 
 <a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_2"><span class="lab"><font color="Green" title="E67">A14</font></span></a>, <a class="ref" href="clvect_1.html#T122" title="CLVECT_1:th.122">CLVECT_1:122</a></span>
<br/>.= 
 <a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:101_2"><span class="lab"><font color="Green" title="E69">A16</font></span></a>, <a class="ref" href="ncfcont1.html#T44" target="_self" title="NCFCONT1:th.44">Th44</a></span>
;<br/><span class="kw">hence </span><a NAME="E9:101_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font> is  <a href="clvect_1.html#V9" title="CLVECT_1:attr.9">convergent</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:101_2"><span class="lab"><font color="Green" title="E69">A16</font></span></a>, <a class="txt" href="#E5:101_2"><span class="lab"><font color="Green" title="E68">A15</font></span></a>, <a class="ref" href="ncfcont1.html#T44" target="_self" title="NCFCONT1:th.44">Th44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:101"/>
<font color="Maroon" title="c4">f1</font> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">-</a> <font color="Maroon" title="c5">f2</font> <a href="ncfcont1.html#R7" title="NCFCONT1:pred.7">is_continuous_on</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101"><span class="lab"><font color="Green" title="E63">A10</font></span></a>, <a class="ref" href="ncfcont1.html#T62" target="_self" title="NCFCONT1:th.62">Th62</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
