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

<span class="kw">set </span><font color="Maroon" title="c3">R</font> = <font color="Maroon" title="c2">cf</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">L</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>
<a NAME="E2:57"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
 <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 
;<br/>
<a NAME="E3:57"/><span class="kw">then </span>
 for <font color="Olive" title="b1">b</font> being   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a> holds  <font color="Maroon" title="c4">L</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <font color="Olive" title="b1">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T6" title="PARTFUN2:th.6">PARTFUN2:6</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">L</font> = <font color="Maroon" title="c4">L</font> as   <a href="cfdiff_1.html#NM2" title="CFDIFF_1:NM.2">C_LINEAR</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="cfdiff_1.html#D4" target="_self" title="CFDIFF_1:def.4">Def4</a></span>;<br/>
<a NAME="E5:57"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">cf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 
<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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">h</font> be   <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">convergent_to_0</a>  <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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">cf</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">h</font></span>)</span> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">cf</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">cf</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">h</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> </span><br/><a NAME="E1:57_1_1"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
( <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">h</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="c2">cf</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:57"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">cf</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">h</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">cf</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">h</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">cf</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1_1"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="funct_2.html#T109" title="FUNCT_2:th.109">FUNCT_2:109</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
<br/>.= 
 <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E2:57_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">cf</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">h</font></span>)</span> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">cf</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">h</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D18" title="VALUED_0:def.18">VALUED_0:def 18</a></span>;<br/><span class="kw">hence </span><a NAME="E3:57_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">cf</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">h</font></span>)</span> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">cf</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cfcont_1.html#T26" title="CFCONT_1:th.26">CFCONT_1:26</a>, <a class="ref" href="cfcont_1.html#T27" title="CFCONT_1:th.27">CFCONT_1:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">R</font> = <font color="Maroon" title="c2">cf</font> as   <a href="cfdiff_1.html#NM1" title="CFDIFF_1:NM.1">C_REST</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="cfdiff_1.html#D3" target="_self" title="CFDIFF_1:def.3">Def3</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Z</font> being   <a href="cfdiff_1.html#V6" title="CFDIFF_1:attr.6">open</a>  <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>  st <font color="Olive" title="b1">Z</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="c7">f</font> &amp; <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Olive" title="b1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Olive" title="b1">Z</font> holds <br/>( <font color="Maroon" title="c7">f</font> <a href="cfdiff_1.html#R2" title="CFDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b1">Z</font> &amp; (  for <font color="Olive" title="b2">x</font> being   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="cfdiff_1.html#K3" title="CFDIFF_1:func.3">`|</a> <font color="Olive" title="b1">Z</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">Z</font> be   <a href="cfdiff_1.html#V6" title="CFDIFF_1:attr.6">open</a>  <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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">Z</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="c7">f</font> &amp; <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c8">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c8">Z</font> implies ( <font color="Maroon" title="c7">f</font> <a href="cfdiff_1.html#R2" title="CFDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c8">Z</font> &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="cfdiff_1.html#K3" title="CFDIFF_1:func.3">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  ) ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E8:57"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
<font color="Maroon" title="c8">Z</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="c7">f</font>
 <span class="kw">and </span><br/><a NAME="E9:57"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c8">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c8">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">f</font> <a href="cfdiff_1.html#R2" title="CFDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c8">Z</font> &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="cfdiff_1.html#K3" title="CFDIFF_1:func.3">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  ) )</span><br/>

<div><span class="lab"><font color="Green" title="E36">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> implies <font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:57_2"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font></span><br/><a NAME="E2:57_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</a></span>;<br/><a NAME="E3:57_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_2"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>;<br/><span class="kw">hence </span><a NAME="E4:57_2"/>
<font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E1:57_2"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E37">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x0</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> implies <font color="Maroon" title="c7">f</font> <a href="cfdiff_1.html#R1" title="CFDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c9">x0</font> )</span><br/><span class="kw">assume </span><a NAME="E1:57_3"/><span class="lab"><font color="Green" title="E38">A9</font></span>: 
<font color="Maroon" title="c9">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">f</font> <a href="cfdiff_1.html#R1" title="CFDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c9">x0</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">N</font> being    <a href="cfdiff_1.html#M1" title="CFDIFF_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c9">x0</font><span class="kw"> such that </span><br/><a NAME="E3:57_3"/><span class="lab"><font color="Green" title="E39">A10</font></span>: 
<font color="Maroon" title="c10">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cfdiff_1.html#T9" target="_self" title="CFDIFF_1:th.9">Th9</a></span>;<br/><a NAME="E4:57_3"/><span class="lab"><font color="Green" title="E40">A11</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c9">x0</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c9">x0</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c9">x0</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c9">x0</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:57_3_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c9">x0</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c9">x0</font></span>)</span></span>)</span></span><br/>

<span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x0</font></span>)</span> = 
<font color="Maroon" title="c11">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:57"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="txt" href="#E3:57_3"><span class="lab"><font color="Green" title="E39">A10</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c11">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c9">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:57"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="txt" href="#E1:57_3"><span class="lab"><font color="Green" title="E38">A9</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c9">x0</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="partfun2.html#T6" title="PARTFUN2:th.6">PARTFUN2:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c9">x0</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c9">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:57_3"/>
<font color="Maroon" title="c10">N</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="c7">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E3:57_3"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:57_3"/>
<font color="Maroon" title="c7">f</font> <a href="cfdiff_1.html#R1" title="CFDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c9">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_3"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="ref" href="cfdiff_1.html#D6" target="_self" title="CFDIFF_1:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E12:57"/><span class="lab"><font color="Green" title="E38">A12</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="cfdiff_1.html#R2" title="CFDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c8">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="cfdiff_1.html#T15" target="_self" title="CFDIFF_1:th.15">Th15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="cfdiff_1.html#K3" title="CFDIFF_1:func.3">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">x0</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="cfdiff_1.html#K3" title="CFDIFF_1:func.3">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E13:57"/><span class="lab"><font color="Green" title="E39">A13</font></span>: 
<font color="Maroon" title="c9">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="cfdiff_1.html#K3" title="CFDIFF_1:func.3">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">N1</font> being    <a href="cfdiff_1.html#M1" title="CFDIFF_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c9">x0</font><span class="kw"> such that </span><br/><a NAME="E15:57"/><span class="lab"><font color="Green" title="E40">A14</font></span>: 
<font color="Maroon" title="c10">N1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cfdiff_1.html#T9" target="_self" title="CFDIFF_1:th.9">Th9</a></span>;<br/>
<a NAME="E16:57"/><span class="lab"><font color="Green" title="E41">A15</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="cfdiff_1.html#R1" title="CFDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c9">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:57"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E13:57"><span class="lab"><font color="Green" title="E39">A13</font></span></a></span>;<br/>
<a NAME="E17:57"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">N</font> being    <a href="cfdiff_1.html#M1" title="CFDIFF_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c9">x0</font> st <br/>( <font color="Olive" title="b1">N</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="c7">f</font> &amp;  ex <font color="Olive" title="b2">L</font> being   <a href="cfdiff_1.html#NM2" title="CFDIFF_1:NM.2">C_LINEAR</a> ex <font color="Olive" title="b3">R</font> being   <a href="cfdiff_1.html#NM1" title="CFDIFF_1:NM.1">C_REST</a> st <br/> for <font color="Olive" title="b4">x</font> being   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">L</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c9">x0</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c9">x0</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfdiff_1.html#D6" target="_self" title="CFDIFF_1:def.6">Def6</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">N</font> being    <a href="cfdiff_1.html#M1" title="CFDIFF_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c9">x0</font><span class="kw"> such that </span><br/><a NAME="E19:57"/><span class="lab"><font color="Green" title="E42">A16</font></span>: 
<font color="Maroon" title="c11">N</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="c7">f</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">N2</font> being    <a href="cfdiff_1.html#M1" title="CFDIFF_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c9">x0</font><span class="kw"> such that </span><br/><a NAME="E21:57"/><span class="lab"><font color="Green" title="E43">A17</font></span>: 
<font color="Maroon" title="c12">N2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">N1</font>
 <span class="kw">and </span><br/><a NAME="E22:57"/><span class="lab"><font color="Green" title="E44">A18</font></span>: 
<font color="Maroon" title="c12">N2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="cfdiff_1.html#E15"><span class="lab"><font color="Green" title="E10">Lm1</font></span></a></span>;<br/>
<a NAME="E23:57"/><span class="lab"><font color="Green" title="E45">A19</font></span>: 
<font color="Maroon" title="c12">N2</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="c7">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:57"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="txt" href="#E22:57"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E24:57"/><span class="lab"><font color="Green" title="E46">A20</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c9">x0</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c9">x0</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N2</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c9">x0</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c9">x0</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:57_4"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c9">x0</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c9">x0</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:57_4"/><span class="kw">then </span>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:57"><span class="lab"><font color="Green" title="E43">A17</font></span></a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x0</font></span>)</span> = 
<font color="Maroon" title="c13">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:57"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="txt" href="#E15:57"><span class="lab"><font color="Green" title="E40">A14</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c13">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c9">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:57"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="txt" href="#E13:57"><span class="lab"><font color="Green" title="E39">A13</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c9">x0</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="partfun2.html#T6" title="PARTFUN2:th.6">PARTFUN2:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c9">x0</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c9">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="cfdiff_1.html#K3" title="CFDIFF_1:func.3">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x0</font> = 
 <a href="cfdiff_1.html#K2" title="CFDIFF_1:func.2">diff</a> (<font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c9">x0</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:57"><span class="lab"><font color="Green" title="E38">A12</font></span></a>, <a class="txt" href="#E13:57"><span class="lab"><font color="Green" title="E39">A13</font></span></a>, <a class="ref" href="cfdiff_1.html#D12" target="_self" title="CFDIFF_1:def.12">Def12</a></span>
<br/>.= 
<font color="Maroon" title="c5">L</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:57"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E23:57"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E24:57"><span class="lab"><font color="Green" title="E46">A20</font></span></a>, <a class="ref" href="cfdiff_1.html#D7" target="_self" title="CFDIFF_1:def.7">Def7</a></span>
<br/>.= 
 <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="partfun2.html#T6" title="PARTFUN2:th.6">PARTFUN2:6</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
