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<b>let </b><font color="Maroon" title="c1">x0</font> be    <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> ; <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">f1</font>, <font color="Olive" title="b2">f2</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">f1</font> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Olive" title="b1">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">f2</font> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c1">x0</font> holds <br/><font color="Olive" title="b2">f2</font> <a href="cfunct_1.html#K1" title="CFUNCT_1:func.1">/</a> <font color="Olive" title="b1">f1</font> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c1">x0</font></span><br/><b>let </b><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">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"> ( <font color="Maroon" title="c2">f1</font> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c2">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">f2</font> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c1">x0</font> implies <font color="Maroon" title="c3">f2</font> <a href="cfunct_1.html#K1" title="CFUNCT_1:func.1">/</a> <font color="Maroon" title="c2">f1</font> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c1">x0</font> )</span><br/>





<b>assume </b><a NAME="E1:60"/>
( <font color="Maroon" title="c2">f1</font> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c2">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">f2</font> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c1">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">f2</font> <a href="cfunct_1.html#K1" title="CFUNCT_1:func.1">/</a> <font color="Maroon" title="c2">f1</font> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c1">x0</font></span><br/>

<a NAME="E2:60"/><b>then </b>
( <font color="Maroon" title="c2">f1</font> <a href="cfunct_1.html#K2" title="CFUNCT_1:func.2">^</a>  <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c1">x0</font> )
 
<b>by </b><i><a class="ref" href="cfcont_1.html#T58" target="_self" title="CFCONT_1:th.58">Th58</a></i>;<br/>
<a NAME="E3:60"/><b>then </b>
<font color="Maroon" title="c3">f2</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cfunct_1.html#K2" title="CFUNCT_1:func.2">^</a> </span>)</span> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c1">x0</font>
 
<b>by </b><i><a class="ref" href="cfcont_1.html#T55" target="_self" title="CFCONT_1:th.55">Th55</a></i>;<br/>
<b>hence </b><a NAME="E4:60"/>
<font color="Maroon" title="c3">f2</font> <a href="cfunct_1.html#K1" title="CFUNCT_1:func.1">/</a> <font color="Maroon" title="c2">f1</font> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="ref" href="cfunct_1.html#T51" title="CFUNCT_1:th.51">CFUNCT_1:51</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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