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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>   st ( for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">x</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="Olive" title="b1">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> holds <br/><font color="Olive" title="b1">f</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Olive" title="b2">x</font> is <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> ) holds <br/>(  <a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Re</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Olive" title="b1">f</font></span>)</span> &amp;  <a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K14" title="MESFUN7C:func.14">Im</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Olive" title="b1">f</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c2">f</font> be  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</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="c2">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> holds <br/><font color="Maroon" title="c2">f</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Olive" title="b1">x</font> is <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> ) implies (  <a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Re</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp;  <a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K14" title="MESFUN7C:func.14">Im</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span> ) )</span><br/>



<b>assume </b><a NAME="E1:41"/><i><font color="Green" title="E23">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</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="c2">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> holds <br/><font color="Maroon" title="c2">f</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Olive" title="b1">x</font> is <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Re</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp;  <a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K14" title="MESFUN7C:func.14">Im</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span> )</span><br/>

<a NAME="E2:41"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Re</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Re</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc8.html#D10" title="MESFUNC8:def.10">MESFUNC8:def 10</a></i>;<br/>
<a NAME="E3:41"/><b>then </b><i><font color="Green" title="E24">P1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Re</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="mesfun7c.html#D11" target="_self" title="MESFUN7C:def.11">Def13</a></i>;<br/>
<a NAME="E4:41"/><i><font color="Green" title="E25">P2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfun6c.html#D1" title="MESFUN6C:def.1">MESFUN6C:def 1</a></i>;<br/>
<a NAME="E5:41"/><b>then </b><i><font color="Green" title="E26">B1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Re</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:41"><i><font color="Green" title="E24">P1</font></i></a>, <a class="ref" href="mesfun7c.html#D10" target="_self" title="MESFUN7C:def.10">Def12</a></i>;<br/>
<a NAME="E6:41"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K14" title="MESFUN7C:func.14">Im</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun7c.html#K14" title="MESFUN7C:func.14">Im</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc8.html#D10" title="MESFUNC8:def.10">MESFUNC8:def 10</a></i>;<br/>
<a NAME="E7:41"/><b>then </b><i><font color="Green" title="E27">Q1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K14" title="MESFUN7C:func.14">Im</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="mesfun7c.html#D12" target="_self" title="MESFUN7C:def.12">Def14</a></i>;<br/>
<a NAME="E8:41"/><i><font color="Green" title="E28">Q2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfun6c.html#D2" title="MESFUN6C:def.2">MESFUN6C:def 2</a></i>;<br/>
<a NAME="E9:41"/><b>then </b><i><font color="Green" title="E29">B2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K14" title="MESFUN7C:func.14">Im</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E7:41"><i><font color="Green" title="E27">Q1</font></i></a>, <a class="ref" href="mesfun7c.html#D10" target="_self" title="MESFUN7C:def.10">Def12</a></i>;<br/>
<div><i><font color="Green" title="E30">P00</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</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"><a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Re</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Re</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> )</span><br/><b>assume </b><a NAME="E1:41_1"/><i><font color="Green" title="E31">P01</font></i>: 
<font color="Maroon" title="c3">x</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"><a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Re</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Re</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span><br/><a NAME="E2:41_1"/><b>then </b><i><font color="Green" title="E32">C1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c3">x</font> is <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>
 <b>by </b><i><a class="txt" href="#E1:41"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="#E3:41"><i><font color="Green" title="E24">P1</font></i></a></i>;<br/><a NAME="E3:41_1"/><b>then </b>
 <a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c3">x</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 ;<br/><a NAME="E4:41_1"/><b>then </b><i><font color="Green" title="E33">PP</font></i>: 
<span class="p1">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Re</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c3">x</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="#E1:41_1"><i><font color="Green" title="E31">P01</font></i></a>, <a class="txt" href="#E3:41"><i><font color="Green" title="E24">P1</font></i></a>, <a class="ref" href="mesfun7c.html#T23" target="_self" title="MESFUN7C:th.23">Th7a</a></i>;<br/><span class="p1">(<span class="default"><a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Re</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> = 
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Re</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:41_1"><i><font color="Green" title="E31">P01</font></i></a>, <a class="ref" href="mesfun7c.html#T14" target="_self" title="MESFUN7C:th.14">Def10a</a></i>
<br/>.= 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Re</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:41_1"><i><font color="Green" title="E33">PP</font></i></a>, <a class="ref" href="rinfsup2.html#T14" title="RINFSUP2:th.14">RINFSUP2:14</a></i>
;<br/><a NAME="E6:41_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Re</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:41_1"><i><font color="Green" title="E31">P01</font></i></a>, <a class="txt" href="#E3:41"><i><font color="Green" title="E24">P1</font></i></a>, <a class="ref" href="mesfun7c.html#T23" target="_self" title="MESFUN7C:th.23">Th7a</a></i>;<br/><a NAME="E7:41_1"/><b>then </b><i><font color="Green" title="E34">P03</font></i>: 
<span class="p1">(<span class="default"><a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Re</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:41_1"><i><font color="Green" title="E32">C1</font></i></a>, <a class="ref" href="comseq_3.html#T41" title="COMSEQ_3:th.41">COMSEQ_3:41</a></i>;<br/><a NAME="E8:41_1"/>
<span class="p1">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:41_1"><i><font color="Green" title="E31">P01</font></i></a>, <a class="txt" href="#E5:41"><i><font color="Green" title="E26">B1</font></i></a>, <a class="ref" href="mesfun6c.html#D1" title="MESFUN6C:def.1">MESFUN6C:def 1</a></i>;<br/><b>hence </b><a NAME="E9:41_1"/>
<span class="p1">(<span class="default"><a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Re</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="#E7:41_1"><i><font color="Green" title="E34">P03</font></i></a>, <a class="txt" href="#E1:41_1"><i><font color="Green" title="E31">P01</font></i></a>, <a class="txt" href="#E5:41"><i><font color="Green" title="E26">B1</font></i></a>, <a class="txt" href="#E4:41"><i><font color="Green" title="E25">P2</font></i></a>, <a class="ref" href="mesfun7c.html#D10" target="_self" title="MESFUN7C:def.10">Def12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E11:41"/>
 <a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> 
 
<b>by </b><i><a class="ref" href="numbers.html#T31" title="NUMBERS:th.31">NUMBERS:31</a>, <a class="ref" href="relset_1.html#T17" title="RELSET_1:th.17">RELSET_1:17</a></i>;<br/>
<b>hence </b><a NAME="E12:41"/>
 <a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Re</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfun6c.html#K1" title="MESFUN6C:func.1">Re</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:41"><i><font color="Green" title="E26">B1</font></i></a>, <a class="txt" href="#E10:41"><i><font color="Green" title="E30">P00</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K14" title="MESFUN7C:func.14">Im</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span></span><br/>

<div><i><font color="Green" title="E31">P00</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</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"><a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K14" title="MESFUN7C:func.14">Im</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K14" title="MESFUN7C:func.14">Im</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> )</span><br/><b>assume </b><a NAME="E1:41_2"/><i><font color="Green" title="E32">P01</font></i>: 
<font color="Maroon" title="c3">x</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"><a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K14" title="MESFUN7C:func.14">Im</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K14" title="MESFUN7C:func.14">Im</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span><br/><a NAME="E2:41_2"/><b>then </b><i><font color="Green" title="E33">D1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c3">x</font> is <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>
 <b>by </b><i><a class="txt" href="#E1:41"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="#E7:41"><i><font color="Green" title="E27">Q1</font></i></a></i>;<br/><a NAME="E3:41_2"/><b>then </b>
 <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c3">x</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 ;<br/><a NAME="E4:41_2"/><b>then </b><i><font color="Green" title="E34">PP</font></i>: 
<span class="p1">(<span class="default"><a href="mesfun7c.html#K14" title="MESFUN7C:func.14">Im</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c3">x</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="#E1:41_2"><i><font color="Green" title="E32">P01</font></i></a>, <a class="txt" href="#E7:41"><i><font color="Green" title="E27">Q1</font></i></a>, <a class="ref" href="mesfun7c.html#T23" target="_self" title="MESFUN7C:th.23">Th7a</a></i>;<br/><span class="p1">(<span class="default"><a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K14" title="MESFUN7C:func.14">Im</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> = 
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfun7c.html#K14" title="MESFUN7C:func.14">Im</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:41_2"><i><font color="Green" title="E32">P01</font></i></a>, <a class="ref" href="mesfun7c.html#T14" target="_self" title="MESFUN7C:th.14">Def10a</a></i>
<br/>.= 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun7c.html#K14" title="MESFUN7C:func.14">Im</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:41_2"><i><font color="Green" title="E34">PP</font></i></a>, <a class="ref" href="rinfsup2.html#T14" title="RINFSUP2:th.14">RINFSUP2:14</a></i>
;<br/><a NAME="E6:41_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K14" title="MESFUN7C:func.14">Im</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:41_2"><i><font color="Green" title="E32">P01</font></i></a>, <a class="txt" href="#E7:41"><i><font color="Green" title="E27">Q1</font></i></a>, <a class="ref" href="mesfun7c.html#T23" target="_self" title="MESFUN7C:th.23">Th7a</a></i>;<br/><a NAME="E7:41_2"/><b>then </b><i><font color="Green" title="E35">P03</font></i>: 
<span class="p1">(<span class="default"><a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K14" title="MESFUN7C:func.14">Im</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:41_2"><i><font color="Green" title="E33">D1</font></i></a>, <a class="ref" href="comseq_3.html#T41" title="COMSEQ_3:th.41">COMSEQ_3:41</a></i>;<br/><a NAME="E8:41_2"/>
<span class="p1">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:41_2"><i><font color="Green" title="E32">P01</font></i></a>, <a class="txt" href="#E9:41"><i><font color="Green" title="E29">B2</font></i></a>, <a class="ref" href="mesfun6c.html#D2" title="MESFUN6C:def.2">MESFUN6C:def 2</a></i>;<br/><b>hence </b><a NAME="E9:41_2"/>
<span class="p1">(<span class="default"><a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K14" title="MESFUN7C:func.14">Im</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="#E7:41_2"><i><font color="Green" title="E35">P03</font></i></a>, <a class="txt" href="#E1:41_2"><i><font color="Green" title="E32">P01</font></i></a>, <a class="txt" href="#E9:41"><i><font color="Green" title="E29">B2</font></i></a>, <a class="txt" href="#E8:41"><i><font color="Green" title="E28">Q2</font></i></a>, <a class="ref" href="mesfun7c.html#D10" target="_self" title="MESFUN7C:def.10">Def12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E14:41"/>
 <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> 
 
<b>by </b><i><a class="ref" href="numbers.html#T31" title="NUMBERS:th.31">NUMBERS:31</a>, <a class="ref" href="relset_1.html#T17" title="RELSET_1:th.17">RELSET_1:17</a></i>;<br/>
<b>hence </b><a NAME="E15:41"/>
 <a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K14" title="MESFUN7C:func.14">Im</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">Im</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E9:41"><i><font color="Green" title="E29">B2</font></i></a>, <a class="txt" href="#E13:41"><i><font color="Green" title="E31">P00</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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