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

<span class="kw">let </span><font color="Maroon" title="c3">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</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">z0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <br/> for <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">y</font> where <font color="Olive" title="b3">y</font> is   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a> : <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">y</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Olive" title="b1">z0</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">r</font> </span>}</span>  holds <br/><font color="Maroon" title="c3">X</font> is  <a href="cfdiff_1.html#V5" title="CFDIFF_1:attr.5">closed</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">z0</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</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">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">y</font> where <font color="Olive" title="b2">y</font> is   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a> : <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">y</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">z0</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> </span>}</span>  holds <br/><font color="Maroon" title="c3">X</font> is  <a href="cfdiff_1.html#V5" title="CFDIFF_1:attr.5">closed</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">r</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <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="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a> : <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">z0</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> </span>}</span>  implies <font color="Maroon" title="c3">X</font> is  <a href="cfdiff_1.html#V5" title="CFDIFF_1:attr.5">closed</a>  )</span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c6">X0</font> = <font color="Maroon" title="c3">X</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  ;<br/>
<span class="kw">assume </span><a NAME="E2:40"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a> : <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">z0</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> </span>}</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="c3">X</font> is  <a href="cfdiff_1.html#V5" title="CFDIFF_1:attr.5">closed</a> </span><br/>

<a NAME="E3:40"/>
for <font color="Olive" title="b1">s1</font> being  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X0</font> &amp; <font color="Olive" title="b1">s1</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  holds <br/> <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Olive" title="b1">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c7">s4</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c5">r</font> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">s2</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c4">z0</font> as   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">s1</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>; <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="c9">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X0</font> &amp; <font color="Maroon" title="c9">s1</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  implies  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c9">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X0</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:40_1"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X0</font>
 <span class="kw">and </span><br/><a NAME="E4:40_1"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
<font color="Maroon" title="c9">s1</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c9">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X0</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c10">s3</font> = <font color="Maroon" title="c9">s1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c8">s2</font>;<br/>
<a NAME="E5:40_1"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<font color="Maroon" title="c8">s2</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfcont_1.html#T48" title="CFCONT_1:th.48">CFCONT_1:48</a></span>;<br/>
<a NAME="E6:40_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A5</font></span>: 
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">s1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c9">s1</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c8">s2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="comseq_2.html#T26" title="COMSEQ_2:th.26">COMSEQ_2:26</a></span>;<br/>
<a NAME="E7:40_1"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
for <font color="Olive" title="b1">n</font> being   <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>  holds  <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">s1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c8">s2</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">s1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c8">s2</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">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> ; <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="c9">s1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">s1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">z0</font></span><br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">s1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">s1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">s1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T8" title="VALUED_1:th.8">VALUED_1:8</a></span>
;<br/><span class="kw">hence </span><a NAME="E2:40_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">s1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">s1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">z0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</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>
<a NAME="E2:40_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">s1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">s1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">z0</font>
 
;<br/>
<a NAME="E3:40_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E4:40_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">s1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E5:40_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">s1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a></span>;<br/>
<a NAME="E6:40_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">y</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">s1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font> &amp; <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">z0</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E21">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:40_1_1"/>
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">s1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c8">s2</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_1"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="valued_1.html#T18" title="VALUED_1:th.18">VALUED_1:18</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>
<a NAME="E8:40_1"/>
<font color="Maroon" title="c9">s1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c8">s2</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E5:40_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="comseq_2.html#T25" title="COMSEQ_2:th.25">COMSEQ_2:25</a></span>;<br/>
<a NAME="E9:40_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A8</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">s1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c8">s2</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">s1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c8">s2</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_2.html#T11" title="COMSEQ_2:th.11">COMSEQ_2:11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">s3</font> = <font color="Maroon" title="c9">s1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c8">s2</font> as   <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E5:40_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="comseq_2.html#T25" title="COMSEQ_2:th.25">COMSEQ_2:25</a></span>;<br/>
<a NAME="E11:40_1"/><span class="lab"><font color="Green" title="E28">A9</font></span>: 
for <font color="Olive" title="b1">n</font> being   <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>  holds  <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c11">s3</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">s4</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c11">s3</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">s4</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span><br/>

<a NAME="E1:40_1_2"/>
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c11">s3</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:40_1_2"/>
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c11">s3</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">s4</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</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>
<a NAME="E12:40_1"/><span class="lab"><font color="Green" title="E29">A10</font></span>: 
for <font color="Olive" title="b1">n</font> being   <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>  holds   <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c8">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c8">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z0</font></span><br/>

 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c8">s2</font> = 
<font color="Maroon" title="c8">s2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfcont_1.html#T50" title="CFCONT_1:th.50">CFCONT_1:50</a></span>
<br/>.= 
<font color="Maroon" title="c4">z0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:40_1_3"/>
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c8">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z0</font>
 ; <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>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c7">s4</font> = 
<font color="Maroon" title="c7">s4</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T41" title="SEQ_4:th.41">SEQ_4:41</a></span>
<br/>.= 
<font color="Maroon" title="c5">r</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/>
<a NAME="E14:40_1"/><span class="kw">then </span>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c11">s3</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:40_1"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="ref" href="seq_2.html#T32" title="SEQ_2:th.32">SEQ_2:32</a></span>;<br/>
<a NAME="E15:40_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c9">s1</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">z0</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:40_1"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="txt" href="#E6:40_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E9:40_1"><span class="lab"><font color="Green" title="E27">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:40_1"/>
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c9">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E21">A1</font></span></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="E4:40"/>
<font color="Maroon" title="c3">X</font> is  <a href="cfdiff_1.html#V5" title="CFDIFF_1:attr.5">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cfdiff_1.html#D10" target="_self" title="CFDIFF_1:def.10">Def10</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>
