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<b>let </b><font color="Maroon">CNS</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">RNS</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">CNS</font>,<font color="Maroon">RNS</font>;<br/>







<b>assume </b><a NAME="E1:163"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> <a href="ncfcont1.html#R14" target="_self">is_Lipschitzian_on</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:163"/>
 <a href="vfunct_1.html#K6">-</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="vfunct_1.html#K4">(#)</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="vfunct_1.html#T29">VFUNCT_1:29</a></i>;<br/>
<b>hence </b><a NAME="E3:163"/>
 <a href="vfunct_1.html#K6">-</a> <font color="Maroon">f</font> <a href="ncfcont1.html#R14" target="_self">is_Lipschitzian_on</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>

<a NAME="E4:163"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:163"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="ncfcont1.html#K3" target="_self">.||</a></span>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="ncfcont1.html#D31" target="_self">NCFCONT1:def 31</a><br/>

<b>consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:163"/><i><font color="Green">E54</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> &amp; ( for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS</font>  st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x2</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">x2</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> ) )
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">s</font>
;<br/>

<b>thus </b><a NAME="E8:163"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">x1</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS</font>, <font color="Maroon">x2</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS</font>;<br/>

<b>assume </b><a NAME="E9:163"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/>

<b>then </b> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font></span>)</span></span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K5">-</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span></span>)</span>
<b>by </b><i>, , </i>
;<br/>
<a NAME="E11:163"/><b>then </b><i><font color="Green">E109</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T13">NORMSP_1:13</a></i>;<br/>
<a NAME="E12:163"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x2</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E13:163"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="ncfcont1.html#K3" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x2</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div>
