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







<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E18">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="c1">X</font>
 ;<br/>

<b>hence </b><a NAME="E2:31"/>
<font color="Maroon" title="c1">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="ncfcont1.html#D27" title="NCFCONT1:def.27">NCFCONT1:def 27</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="ncfcont2.html#D1" target="_self" title="NCFCONT2:def.1">NCFCONT2:def 1</a><br/>

<b>consider </b><font color="Maroon" title="c5">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:31"/><i><font color="Green" title="E19">A2</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</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="c2">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="c1">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="c1">X</font> holds <br/><span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">x1</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">x2</font></span>)</span></span>)</span></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> <font color="Maroon" title="c5">r</font> <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="Olive" title="b1">x1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Olive" title="b2">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="ncfcont2.html#E1:31"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="ncfcont1.html#D27" title="NCFCONT1:def.27">NCFCONT1:def 27</a></i>;<br/>
<b>let </b><font color="Maroon" title="c6">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E5:31"/><i><font color="Green" title="E20">A3</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">p</font>
 ;<br/>

<b>take </b><font color="Maroon" title="c7">s</font> = <font color="Maroon" title="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font>;<br/>

<b>thus </b><a NAME="E6:31"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">s</font>
 <b>by </b><i><a class="txt" href="ncfcont2.html#E4:31"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="ncfcont2.html#E5:31"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></i>;<br/>

<b>let </b><font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c9">x2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">CNS1</font>;<br/>

<b>assume </b><a NAME="E7:31"/><i><font color="Green" title="E21">A4</font></i>: 
( <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c9">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <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="c8">x1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c9">x2</font></span>)</span></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> <font color="Maroon" title="c7">s</font> )
 ;<br/>

<a NAME="E8:31"/><b>then </b><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c5">r</font> <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="c8">x1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c9">x2</font></span>)</span></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> <font color="Maroon" title="c7">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="txt" href="ncfcont2.html#E4:31"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E9:31"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">x1</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">x2</font></span>)</span></span>)</span></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> <font color="Maroon" title="c5">r</font> <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="c8">x1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c9">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="ncfcont2.html#E4:31"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="ncfcont2.html#E7:31"><i><font color="Green" title="E21">A4</font></i></a></i>;<br/>
<a NAME="E10:31"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">x1</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">x2</font></span>)</span></span>)</span></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">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="txt" href="ncfcont2.html#E8:31"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E11:31"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">x1</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">x2</font></span>)</span></span>)</span></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> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="txt" href="ncfcont2.html#E4:31"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/>


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