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<b>let </b><font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">f2</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x0</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st <font color="Maroon" title="c3">f1</font> <a href="cfdiff_1.html#R1" title="CFDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">x0</font> &amp; <font color="Maroon" title="c4">f2</font> <a href="cfdiff_1.html#R1" title="CFDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">x0</font> holds <br/>( <font color="Maroon" title="c3">f1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">f2</font> <a href="cfdiff_1.html#R1" title="CFDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">x0</font> &amp;  <a href="cfdiff_1.html#K2" title="CFDIFF_1:func.2">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Olive" title="b1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cfdiff_1.html#K2" title="CFDIFF_1:func.2">diff</a> <font color="Maroon" title="c3">f1</font>,<font color="Olive" title="b1">x0</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><a href="cfdiff_1.html#K2" title="CFDIFF_1:func.2">diff</a> <font color="Maroon" title="c4">f2</font>,<font color="Olive" title="b1">x0</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c5">x0</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f1</font> <a href="cfdiff_1.html#R1" title="CFDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c5">x0</font> &amp; <font color="Maroon" title="c4">f2</font> <a href="cfdiff_1.html#R1" title="CFDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c5">x0</font> implies ( <font color="Maroon" title="c3">f1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">f2</font> <a href="cfdiff_1.html#R1" title="CFDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c5">x0</font> &amp;  <a href="cfdiff_1.html#K2" title="CFDIFF_1:func.2">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cfdiff_1.html#K2" title="CFDIFF_1:func.2">diff</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">x0</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><a href="cfdiff_1.html#K2" title="CFDIFF_1:func.2">diff</a> <font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c5">x0</font></span>)</span> ) )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:54"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c3">f1</font> <a href="cfdiff_1.html#R1" title="CFDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c5">x0</font>
 <b>and </b><br/><a NAME="E2:54"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c4">f2</font> <a href="cfdiff_1.html#R1" title="CFDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c5">x0</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">f1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">f2</font> <a href="cfdiff_1.html#R1" title="CFDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c5">x0</font> &amp;  <a href="cfdiff_1.html#K2" title="CFDIFF_1:func.2">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cfdiff_1.html#K2" title="CFDIFF_1:func.2">diff</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">x0</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><a href="cfdiff_1.html#K2" title="CFDIFF_1:func.2">diff</a> <font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c5">x0</font></span>)</span> )</span><br/>

<b>consider </b><font color="Maroon" title="c6">N1</font> being    <a href="cfdiff_1.html#M1" title="CFDIFF_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c5">x0</font><b> such that </b><br/><a NAME="E4:54"/><i><font color="Green" title="E32">A3</font></i>: 
( <font color="Maroon" title="c6">N1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f1</font> &amp;  ex <font color="Olive" title="b1">L</font> being  <a href="cfdiff_1.html#NM2" title="CFDIFF_1:NM.2">C_LINEAR</a> ex <font color="Olive" title="b2">R</font> being  <a href="cfdiff_1.html#NM1" title="CFDIFF_1:NM.1">C_REST</a> st <br/>for <font color="Olive" title="b3">x</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">N1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">L</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">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="b2">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:54"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="cfdiff_1.html#D6" target="_self" title="CFDIFF_1:def.6">Def6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">L1</font> being   <a href="cfdiff_1.html#NM2" title="CFDIFF_1:NM.2">C_LINEAR</a>, <font color="Maroon" title="c8">R1</font> being   <a href="cfdiff_1.html#NM1" title="CFDIFF_1:NM.1">C_REST</a><b> such that </b><br/><a NAME="E6:54"/><i><font color="Green" title="E33">A4</font></i>: 
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="c6">N1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</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="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">L1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="c5">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="c8">R1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="c5">x0</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:54"><i><font color="Green" title="E32">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">N2</font> being    <a href="cfdiff_1.html#M1" title="CFDIFF_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c5">x0</font><b> such that </b><br/><a NAME="E8:54"/><i><font color="Green" title="E34">A5</font></i>: 
( <font color="Maroon" title="c9">N2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f2</font> &amp;  ex <font color="Olive" title="b1">L</font> being  <a href="cfdiff_1.html#NM2" title="CFDIFF_1:NM.2">C_LINEAR</a> ex <font color="Olive" title="b2">R</font> being  <a href="cfdiff_1.html#NM1" title="CFDIFF_1:NM.1">C_REST</a> st <br/>for <font color="Olive" title="b3">x</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">N2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">L</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">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="b2">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E2:54"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="cfdiff_1.html#D6" target="_self" title="CFDIFF_1:def.6">Def6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">L2</font> being   <a href="cfdiff_1.html#NM2" title="CFDIFF_1:NM.2">C_LINEAR</a>, <font color="Maroon" title="c11">R2</font> being   <a href="cfdiff_1.html#NM1" title="CFDIFF_1:NM.1">C_REST</a><b> such that </b><br/><a NAME="E10:54"/><i><font color="Green" title="E35">A6</font></i>: 
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="c9">N2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</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="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">L2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="c5">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="c11">R2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="c5">x0</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E8:54"><i><font color="Green" title="E34">A5</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">N</font> being    <a href="cfdiff_1.html#M1" title="CFDIFF_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c5">x0</font><b> such that </b><br/><a NAME="E12:54"/><i><font color="Green" title="E36">A7</font></i>: 
( <font color="Maroon" title="c12">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">N1</font> &amp; <font color="Maroon" title="c12">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">N2</font> )
 <b>by </b><i><a class="txt" href="cfdiff_1.html#E15"><i><font color="Green" title="E10">Th10</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">L</font> = <font color="Maroon" title="c7">L1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c10">L2</font> as   <a href="cfdiff_1.html#NM2" title="CFDIFF_1:NM.2">C_LINEAR</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c14">R</font> = <font color="Maroon" title="c8">R1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c11">R2</font> as   <a href="cfdiff_1.html#NM1" title="CFDIFF_1:NM.1">C_REST</a> ;<br/>
<a NAME="E15:54"/><i><font color="Green" title="E37">A10</font></i>: 
<font color="Maroon" title="c12">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f1</font>
 
<b>by </b><i><a class="txt" href="#E4:54"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="#E12:54"><i><font color="Green" title="E36">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E16:54"/>
<font color="Maroon" title="c12">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f2</font>
 
<b>by </b><i><a class="txt" href="#E8:54"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="#E12:54"><i><font color="Green" title="E36">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E17:54"/><b>then </b>
<font color="Maroon" title="c12">N</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E15:54"><i><font color="Green" title="E37">A10</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E18:54"/><b>then </b><i><font color="Green" title="E38">A11</font></i>: 
<font color="Maroon" title="c12">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="cfunct_1.html#T4" title="CFUNCT_1:th.4">CFUNCT_1:4</a></i>;<br/>
<div><i><font color="Green" title="E39">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">x</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">L</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">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="c14">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> )</span><br/><b>assume </b><a NAME="E1:54_1"/><i><font color="Green" title="E40">A13</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font>
 ; <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"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">L</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">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="c14">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span></span><br/><a NAME="E2:54_1"/><i><font color="Green" title="E41">A14</font></i>: 
<font color="Maroon" title="c5">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font>
 <b>by </b><i><a class="ref" href="cfdiff_1.html#T7" target="_self" title="CFDIFF_1:th.7">Th9</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="#E18:54"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="#E1:54_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="ref" href="cfunct_1.html#T4" title="CFUNCT_1:th.4">CFUNCT_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E18:54"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="#E2:54_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="ref" href="cfunct_1.html#T4" title="CFUNCT_1:th.4">CFUNCT_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">L1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">R1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E6:54"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="#E12:54"><i><font color="Green" title="E36">A7</font></i></a>, <a class="txt" href="#E1:54_1"><i><font color="Green" title="E40">A13</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">L1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">R1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">L2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">R2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E10:54"><i><font color="Green" title="E35">A6</font></i></a>, <a class="txt" href="#E12:54"><i><font color="Green" title="E36">A7</font></i></a>, <a class="txt" href="#E1:54_1"><i><font color="Green" title="E40">A13</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">L1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">L2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">R1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">R2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">L</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">R1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">R2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cfunct_1.html#T76" title="CFUNCT_1:th.76">CFUNCT_1:76</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">L</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">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="c14">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cfunct_1.html#T76" title="CFUNCT_1:th.76">CFUNCT_1:76</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>
<b>hence </b><a NAME="E20:54"/>
<font color="Maroon" title="c3">f1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">f2</font> <a href="cfdiff_1.html#R1" title="CFDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c5">x0</font>
 <b>by </b><i><a class="txt" href="#E18:54"><i><font color="Green" title="E38">A11</font></i></a>, <a class="ref" href="cfdiff_1.html#D6" target="_self" title="CFDIFF_1:def.6">Def6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="cfdiff_1.html#K2" title="CFDIFF_1:func.2">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cfdiff_1.html#K2" title="CFDIFF_1:func.2">diff</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">x0</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><a href="cfdiff_1.html#K2" title="CFDIFF_1:func.2">diff</a> <font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c5">x0</font></span>)</span></span><br/>

<b>hence </b> <a href="cfdiff_1.html#K2" title="CFDIFF_1:func.2">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c5">x0</font> = 
<font color="Maroon" title="c13">L</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> 
<b>by </b><i><a class="txt" href="#E18:54"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="#E19:54"><i><font color="Green" title="E39">A12</font></i></a>, <a class="ref" href="cfdiff_1.html#D7" target="_self" title="CFDIFF_1:def.7">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">L1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">L2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span>
<b>by </b><i><a class="ref" href="cfunct_1.html#T76" title="CFUNCT_1:th.76">CFUNCT_1:76</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cfdiff_1.html#K2" title="CFDIFF_1:func.2">diff</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">x0</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">L2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E1:54"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="#E4:54"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="#E6:54"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="cfdiff_1.html#D7" target="_self" title="CFDIFF_1:def.7">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cfdiff_1.html#K2" title="CFDIFF_1:func.2">diff</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">x0</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><a href="cfdiff_1.html#K2" title="CFDIFF_1:func.2">diff</a> <font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c5">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:54"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="#E8:54"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="#E10:54"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="cfdiff_1.html#D7" target="_self" title="CFDIFF_1:def.7">Def7</a></i>
;<br/> <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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