<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">CNS1</font>, <font color="Maroon" title="c2">CNS2</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c1">CNS1</font>,<font color="Maroon" title="c2">CNS2</font>;<br/>







<b>assume </b><a NAME="E1:160"/><i><font color="Green" title="E116">A1</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="ncfcont1.html#R13" title="NCFCONT1:pred.13">is_Lipschitzian_on</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E2:160"/>
 <a href="vfunct_2.html#K6" title="VFUNCT_2:func.6">-</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span> <a href="vfunct_2.html#K4" title="VFUNCT_2:func.4">(#)</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="vfunct_2.html#T23" title="VFUNCT_2:th.23">VFUNCT_2:23</a></i>;<br/>
<b>hence </b><a NAME="E3:160"/>
 <a href="vfunct_2.html#K6" title="VFUNCT_2:func.6">-</a> <font color="Maroon" title="c4">f</font> <a href="ncfcont1.html#R13" title="NCFCONT1:pred.13">is_Lipschitzian_on</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E1:160"><i><font color="Green" title="E116">A1</font></i></a>, <a class="ref" href="ncfcont1.html#T127" target="_self" title="NCFCONT1:th.127">Th127</a></i>;<br/>

<a NAME="E4:160"/>
<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="txt" href="ncfcont1.html#E1:160"><i><font color="Green" title="E116">A1</font></i></a>, <a class="ref" href="ncfcont1.html#D27" target="_self" title="NCFCONT1:def.27">Def27</a></i>;<br/>
<b>hence </b><a NAME="E5:160"/><i><font color="Green" title="E117">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> <span class="p1"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</a></span>
 <b>by </b><i><a class="ref" href="ncfcont1.html#D2" target="_self" title="NCFCONT1:def.2">Def2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="ncfcont1.html#D31" target="_self" title="NCFCONT1:def.31">NCFCONT1:def 31</a><br/>

<b>consider </b><font color="Maroon" title="c5">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E7:160"/><i><font color="Green" title="E118">A3</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">s</font> &amp; ( for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CNS1</font>  st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/><span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x2</font></span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b2">x2</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> ) )
 <b>by </b><i><a class="txt" href="ncfcont1.html#E1:160"><i><font color="Green" title="E116">A1</font></i></a>, <a class="ref" href="ncfcont1.html#D27" target="_self" title="NCFCONT1:def.27">Def27</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">s</font>
;<br/>

<b>thus </b><a NAME="E8:160"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">s</font>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E7:160"><i><font color="Green" title="E118">A3</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CNS1</font>;<br/>

<b>assume </b><a NAME="E9:160"/><i><font color="Green" title="E119">A4</font></i>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 ;<br/>

<b>then </b> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E5:160"><i><font color="Green" title="E117">A2</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E5:160"><i><font color="Green" title="E117">A2</font></i></a>, <a class="txt" href="ncfcont1.html#E9:160"><i><font color="Green" title="E119">A4</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x1</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E5:160"><i><font color="Green" title="E117">A2</font></i></a>, <a class="txt" href="ncfcont1.html#E9:160"><i><font color="Green" title="E119">A4</font></i></a>, <a class="ref" href="ncfcont1.html#D2" target="_self" title="NCFCONT1:def.2">Def2</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x1</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E5:160"><i><font color="Green" title="E117">A2</font></i></a>, <a class="txt" href="ncfcont1.html#E9:160"><i><font color="Green" title="E119">A4</font></i></a>, <a class="ref" href="ncfcont1.html#D2" target="_self" title="NCFCONT1:def.2">Def2</a></i>
;<br/>
<a NAME="E11:160"/><b>then </b><i><font color="Green" title="E120">A5</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
 
<b>by </b><i><a class="ref" href="clvect_1.html#T111" title="CLVECT_1:th.111">CLVECT_1:111</a></i>;<br/>
<a NAME="E12:160"/>
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
 
<b>by </b><i><a class="txt" href="ncfcont1.html#E7:160"><i><font color="Green" title="E118">A3</font></i></a>, <a class="txt" href="ncfcont1.html#E9:160"><i><font color="Green" title="E119">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:160"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K1" title="NCFCONT1:func.1">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E11:160"><i><font color="Green" title="E120">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div>
