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<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="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:10"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">f</font> <a href="fcont_1.html#R3">is_Lipschitzian_on</a> <font color="Maroon">X</font>
 ;<br/>

<b>hence </b><a NAME="E2:10"/>
<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><a class="ref" href="fcont_1.html#D3">FCONT_1:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="fcont_2.html#D1" target="_self">FCONT_2:def 1</a><br/>

<b>consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:10"/><i><font color="Green">E42</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; ( for <font color="Olive">x1</font>, <font color="Olive">x2</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   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/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">x2</font></span>)</span></span>)</span> ) )
 <b>by </b><i>, <a class="ref" href="fcont_1.html#D3">FCONT_1:def 3</a></i>;<br/>
<b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E5:10"/><i><font color="Green">E43</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>take </b><font color="Maroon">s</font> = <font color="Maroon">p</font> <a href="real_1.html#K6">/</a> <font color="Maroon">r</font>;<br/>

<b>thus </b><a NAME="E7:10"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/>

<b>let </b><font color="Maroon">x1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">x2</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E8:10"/><i><font color="Green">E44</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> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> )
 ;<br/>

<a NAME="E9:10"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K5">-</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> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E10:10"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <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"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:10"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <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"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K6">/</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E12:10"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <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"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>


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