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<b>let </b><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="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>   st <font color="Olive" title="b1">f</font> <a href="cfdiff_1.html#R2" title="CFDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Olive" title="b1">f</font> <a href="cfcont_1.html#R2" title="CFCONT_1:pred.2">is_continuous_on</a> <font color="Maroon" title="c3">X</font></span><br/><b>let </b><font color="Maroon" title="c4">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> <a href="cfdiff_1.html#R2" title="CFDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c4">f</font> <a href="cfcont_1.html#R2" title="CFCONT_1:pred.2">is_continuous_on</a> <font color="Maroon" title="c3">X</font> )</span><br/>



<b>assume </b><a NAME="E1:65"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="cfdiff_1.html#R2" title="CFDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="cfcont_1.html#R2" title="CFCONT_1:pred.2">is_continuous_on</a> <font color="Maroon" title="c3">X</font></span><br/>

<b>hence </b><a NAME="E2:65"/><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c3">X</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">f</font>
 <b>by </b><i><a class="ref" href="cfdiff_1.html#D9" target="_self" title="CFDIFF_1:def.9">Def9</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cfcont_1.html#D5" title="CFCONT_1:def.5">CFCONT_1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> or <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c3">X</font> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>

<b>let </b><font color="Maroon" title="c5">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"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> or <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c3">X</font> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x</font> )</span><br/>

<b>assume </b><a NAME="E3:65"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c3">X</font> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x</font></span><br/>

<a NAME="E4:65"/><b>then </b><i><font color="Green" title="E35">A3</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c3">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:65"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="relat_1.html#T86" title="RELAT_1:th.86">RELAT_1:86</a></i>;<br/>
<a NAME="E5:65"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c3">X</font> is <a href="cfdiff_1.html#V4" title="CFDIFF_1:attr.4">differentiable</a>
 
<b>by </b><i><a class="txt" href="#E1:65"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="cfdiff_1.html#D9" target="_self" title="CFDIFF_1:def.9">Def9</a></i>;<br/>
<a NAME="E6:65"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c3">X</font> <a href="cfdiff_1.html#R1" title="CFDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c5">x</font>
 
<b>by </b><i><a class="txt" href="#E4:65"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="cfdiff_1.html#D8" target="_self" title="CFDIFF_1:def.8">Def8</a></i>;<br/>
<b>hence </b><a NAME="E7:65"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c3">X</font> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="cfdiff_1.html#T34" target="_self" title="CFDIFF_1:th.34">Th38</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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