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<b>let </b><font color="Maroon" title="c1">CNS</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</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="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c1">CNS</font>,<font color="Maroon" title="c1">CNS</font>  st <font color="Olive" title="b1">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">f</font></span>)</span> holds <br/><font color="Olive" title="b1">f</font> <a href="ncfcont1.html#R7" title="NCFCONT1:pred.7">is_continuous_on</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">f</font></span><br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c1">CNS</font>,<font color="Maroon" title="c1">CNS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> implies <font color="Maroon" title="c2">f</font> <a href="ncfcont1.html#R7" title="NCFCONT1:pred.7">is_continuous_on</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> )</span><br/>



<b>assume </b><a NAME="E1:151"/><i><font color="Green" title="E99">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="ncfcont1.html#R7" title="NCFCONT1:pred.7">is_continuous_on</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="151_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CNS</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">x0</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> <font color="Maroon" title="c2">f</font> implies <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x0</font> )</span><br/><b>assume </b><a NAME="E1:151_1"/><i><font color="Green" title="E100">A2</font></i>: 
<font color="Maroon" title="c3">x0</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> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x0</font></span><br/><b>thus </b><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">x0</font> = 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x0</font>
<b>by </b><i><a class="txt" href="#E1:151_1"><i><font color="Green" title="E100">A2</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c3">x0</font>
<b>by </b><i><a class="txt" href="#E1:151"><i><font color="Green" title="E99">A1</font></i></a>, <a class="txt" href="#E1:151_1"><i><font color="Green" title="E100">A2</font></i></a>, <a class="ref" href="funct_1.html#T34" title="FUNCT_1:th.34">FUNCT_1:34</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="E3:151"/>
<font color="Maroon" title="c2">f</font> <a href="ncfcont1.html#R7" title="NCFCONT1:pred.7">is_continuous_on</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="ref" href="ncfcont1.html#T149" target="_self" title="NCFCONT1:th.149">Th149</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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