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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E1:32"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="ncfcont1.html#R14">is_Lipschitzian_on</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:32"/>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="ncfcont1.html#D28">NCFCONT1:def 28</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="ncfcont2.html#D2" target="_self">NCFCONT2:def 2</a><br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:32"/><i><font color="Green">E20</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></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">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></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">c<sub>5</sub></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">b<sub>1</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> ) )
 <b>by </b><i><a class="txt" href="ncfcont2.html#E1:32"><i><font color="Green">E19</font></i></a>, <a class="ref" href="ncfcont1.html#D28">NCFCONT1:def 28</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E5:32"/><i><font color="Green">E21</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>5</sub></font>;<br/>

<b>thus </b><a NAME="E6:32"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="ncfcont2.html#E4:32"><i><font color="Green">E20</font></i></a>, <a class="txt" href="ncfcont2.html#E5:32"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><a NAME="E7:32"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<a NAME="E8:32"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>5</sub></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">c<sub>8</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="ncfcont2.html#E4:32"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E9:32"/>
<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">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></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">c<sub>8</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 
<b>by </b><i><a class="txt" href="ncfcont2.html#E4:32"><i><font color="Green">E20</font></i></a>, <a class="txt" href="ncfcont2.html#E7:32"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E10:32"/><b>then </b>
<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">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="ncfcont2.html#E8:32"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E11:32"/>
<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">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></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">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="ncfcont2.html#E4:32"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>


</div>
