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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:54"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:54"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:54"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<font color="Maroon" title="c6">N1</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="c3">f1</font>
 <span class="kw">and </span><br/><a NAME="E5:54"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
 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="funct_2.html#K10" title="FUNCT_2:func.10">/.</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="funct_2.html#K10" title="FUNCT_2:func.10">/.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="cfdiff_1.html#D6" target="_self" title="CFDIFF_1:def.6">Def6</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E7:54"/><span class="lab"><font color="Green" title="E34">A5</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="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="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="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="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="c5">x0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:54"><span class="lab"><font color="Green" title="E33">A4</font></span></a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E9:54"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
<font color="Maroon" title="c9">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="c4">f2</font>
 <span class="kw">and </span><br/><a NAME="E10:54"/><span class="lab"><font color="Green" title="E36">A7</font></span>: 
 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="funct_2.html#K10" title="FUNCT_2:func.10">/.</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="funct_2.html#K10" title="FUNCT_2:func.10">/.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="cfdiff_1.html#D6" target="_self" title="CFDIFF_1:def.6">Def6</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E12:54"/><span class="lab"><font color="Green" title="E37">A8</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="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="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="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="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="c5">x0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:54"><span class="lab"><font color="Green" title="E36">A7</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">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/>
<span class="kw">reconsider </span><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/>
<span class="kw">consider </span><font color="Maroon" title="c14">N</font> being    <a href="cfdiff_1.html#M1" title="CFDIFF_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c5">x0</font><span class="kw"> such that </span><br/><a NAME="E16:54"/><span class="lab"><font color="Green" title="E38">A9</font></span>: 
<font color="Maroon" title="c14">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">N1</font>
 <span class="kw">and </span><br/><a NAME="E17:54"/><span class="lab"><font color="Green" title="E39">A10</font></span>: 
<font color="Maroon" title="c14">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">N2</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="E18:54"/><span class="lab"><font color="Green" title="E40">A11</font></span>: 
<font color="Maroon" title="c14">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="c4">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:54"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="txt" href="#E17:54"><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/>
<a NAME="E19:54"/>
<font color="Maroon" title="c14">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="c3">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E16:54"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E20:54"/><span class="kw">then </span>
<font color="Maroon" title="c14">N</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c14">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:54"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></span>;<br/>
<a NAME="E21:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A12</font></span>: 
<font color="Maroon" title="c14">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> <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfunct_1.html#T2" title="CFUNCT_1:th.2">CFUNCT_1:2</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E42">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c14">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="funct_2.html#K10" title="FUNCT_2:func.10">/.</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="c12">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</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="E1:54_1"/><span class="lab"><font color="Green" title="E43">A14</font></span>: 
<font color="Maroon" title="c5">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cfdiff_1.html#T7" target="_self" title="CFDIFF_1:th.7">Th7</a></span>;<br/><span class="kw">assume </span><a NAME="E2:54_1"/><span class="lab"><font color="Green" title="E44">A15</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">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"><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="funct_2.html#K10" title="FUNCT_2:func.10">/.</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="c12">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</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/><span class="kw">hence </span><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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:54"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="ref" href="cfunct_1.html#T2" title="CFUNCT_1:th.2">CFUNCT_1:2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:54"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E1:54_1"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="ref" href="cfunct_1.html#T2" title="CFUNCT_1:th.2">CFUNCT_1:2</a></span>
<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="funct_2.html#K10" title="FUNCT_2:func.10">/.</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="funct_2.html#K10" title="FUNCT_2:func.10">/.</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:54"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E16:54"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E44">A15</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">L1</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</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="funct_2.html#K10" title="FUNCT_2:func.10">/.</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="funct_2.html#K10" title="FUNCT_2:func.10">/.</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="funct_2.html#K10" title="FUNCT_2:func.10">/.</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:54"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E17:54"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E44">A15</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">L1</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</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="funct_2.html#K10" title="FUNCT_2:func.10">/.</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="funct_2.html#K10" title="FUNCT_2:func.10">/.</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="funct_2.html#K10" title="FUNCT_2:func.10">/.</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="funct_2.html#K10" title="FUNCT_2:func.10">/.</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="funct_2.html#K10" title="FUNCT_2:func.10">/.</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="funct_2.html#K10" title="FUNCT_2:func.10">/.</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfunct_1.html#T64" title="CFUNCT_1:th.64">CFUNCT_1:64</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">L</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</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="c12">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</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 class="kw">by</span> <span class="lab"><a class="ref" href="cfunct_1.html#T64" title="CFUNCT_1:th.64">CFUNCT_1:64</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">hence </span><a NAME="E23:54"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:54"><span class="lab"><font color="Green" title="E41">A12</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">  <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/>

<span class="kw">hence </span> <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="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="#E21:54"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E22:54"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="ref" href="cfdiff_1.html#D7" target="_self" title="CFDIFF_1:def.7">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">L1</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>)</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="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfunct_1.html#T64" title="CFUNCT_1:th.64">CFUNCT_1:64</a></span>
<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="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E4:54"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E7:54"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="ref" href="cfdiff_1.html#D7" target="_self" title="CFDIFF_1:def.7">Def7</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E9:54"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="txt" href="#E12:54"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="cfdiff_1.html#D7" target="_self" title="CFDIFF_1:def.7">Def7</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/>


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