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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f1</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>let </b><font color="Maroon">f2</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="E1:36"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">f1</font> <a href="fcont_1.html#R3" target="_self">is_Lipschitzian_on</a> <font color="Maroon">X</font> &amp; <font color="Maroon">f2</font> <a href="fcont_1.html#R3" target="_self">is_Lipschitzian_on</a> <font color="Maroon">X1</font> )
 ;<br/>

<a NAME="E2:36"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E3:36"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">f1</font> <a href="fcont_1.html#R3" target="_self">is_Lipschitzian_on</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:36"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:36"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f2</font> <a href="fcont_1.html#R3" target="_self">is_Lipschitzian_on</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X1</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:36"/><b>then </b>
<font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:36"/><b>then </b>
<font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<b>hence </b><a NAME="E8:36"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="fcont_1.html#D3" target="_self">FCONT_1:def 3</a><br/>

<b>consider </b><font color="Maroon">s</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="E10:36"/><i><font color="Green">E45</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</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> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X1</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X1</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">f1</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">f1</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">s</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>, </i>;<br/>
<b>consider </b><font color="Maroon">g</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="E12:36"/><i><font color="Green">E46</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</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> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X1</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X1</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">f2</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">f2</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">g</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>, </i>;<br/>
<b>take </b><font color="Maroon">p</font> = <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font>;<br/>

<a NAME="E13:36"/>
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E14:36"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/>

<b>let </b><font color="Maroon">x1</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">x2</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E15:36"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X1</font> )
 ;<br/>

<a NAME="E16:36"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f2</font></span>)</span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f2</font></span>)</span> )
 
<b>by </b><i/>;<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">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f2</font></span>)</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">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f2</font></span>)</span> <a href="seq_1.html#K2">.</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">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f2</font></span>)</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="seq_1.html#D4">SEQ_1: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">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="seq_1.html#D4">SEQ_1: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">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span>

;<br/>
<a NAME="E18:36"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f2</font></span>)</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">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f2</font></span>)</span> <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"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T143">COMPLEX1:143</a></i>;<br/>
<a NAME="E19:36"/><i><font color="Green">E53</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</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">f1</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">s</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="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E20:36"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</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">f2</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">g</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="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E21:36"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E22:36"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f2</font></span>)</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">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f2</font></span>)</span> <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> <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="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div>
