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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>;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">CNS</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:184"/><i><font color="Green" title="E125">A1</font></i>: 
for <font color="Olive" title="b1">x0</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CNS</font>  st <font color="Olive" title="b1">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> holds <br/><font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Olive" title="b1">x0</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="184_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">x2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CNS</font>;<br/><b>assume </b><a NAME="E1:184_1"/>
( <font color="Maroon" title="c3">x1</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> &amp; <font color="Maroon" title="c4">x2</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> )
 ;<br/><a NAME="E2:184_1"/><b>then </b><i><font color="Green" title="E126">A2</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c3">x1</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> &amp; <font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c4">x2</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> )
 <b>by </b><i><a class="txt" href="ncfcont1.html#E1:184"><i><font color="Green" title="E125">A1</font></i></a></i>;<br/><a NAME="E3:184_1"/><i><font color="Green" title="E127">A3</font></i>: 
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c3">x1</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c4">x2</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">x2</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
 <b>by </b><i><a class="ref" href="clvect_1.html#T110" title="CLVECT_1:th.110">CLVECT_1:110</a></i>;<br/><a NAME="E4:184_1"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c4">x2</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c3">x1</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c3">x1</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
 <b>by </b><i><a class="ref" href="clvect_1.html#T110" title="CLVECT_1:th.110">CLVECT_1:110</a></i>;<br/><a NAME="E5:184_1"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c4">x2</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c3">x1</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">x2</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
 <b>by </b><i><a class="ref" href="clvect_1.html#T109" title="CLVECT_1:th.109">CLVECT_1:109</a></i>;<br/><a NAME="E6:184_1"/><b>then </b>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c3">x1</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c4">x2</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">x2</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/><b>hence </b><a NAME="E7:184_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">x1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">x2</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E2:184_1"><i><font color="Green" title="E126">A2</font></i></a>, <a class="txt" href="ncfcont1.html#E3:184_1"><i><font color="Green" title="E127">A3</font></i></a>, <a class="ref" href="absvalue.html#T12" title="ABSVALUE:th.12">ABSVALUE:12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:184"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="ncfcont1.html#R17" title="NCFCONT1:pred.17">is_Lipschitzian_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#D31" target="_self" title="NCFCONT1:def.31">Def31</a></i>;<br/>
<b>hence </b><a NAME="E4:184"/>
<font color="Maroon" title="c2">f</font> <a href="ncfcont1.html#R11" title="NCFCONT1:pred.11">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#T141" target="_self" title="NCFCONT1:th.141">Th141</a></i>;<br/>


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