<?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">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>thus </b><a NAME="E1:15"/>
( <font color="Maroon">f</font> <a href="fcont_1.html#R2" target="_self">is_continuous_on</a> <font color="Maroon">X</font> implies ( <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> &amp; ( for <font color="Olive">x0</font>, <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> holds <br/> ex <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> &amp; ( for <font color="Olive">x1</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;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">x0</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</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">x0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> ) ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> <a href="fcont_1.html#R2" target="_self">is_continuous_on</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:15_1"/><b>then </b><i><font color="Green">E39</font></i>: 
( <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> &amp; ( for <font color="Olive">x1</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> holds <br/><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font> <a href="fcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Olive">x1</font> ) )
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:15_1"/>
<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>let </b><font color="Maroon">x0</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">r</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="E4:15_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 ;<br/>

<a NAME="E5:15_1"/><b>then </b>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font> <a href="fcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">x0</font>
 
<b>by </b><i>, </i>;<br/>
<b>then 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="E7:15_1"/><i><font color="Green">E42</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> &amp; ( for <font color="Olive">x1</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>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> holds <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">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> ) )
 <b>by </b><i>, </i>;<br/>
<i><font color="Green">E43</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<font color="Maroon">X</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>take </b>
<font color="Maroon">s</font>
;<br/>

<b>thus </b><a NAME="E9:15_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<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>assume </b><a NAME="E10:15_1"/><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;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> )
 ;<br/>

<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">x0</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">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</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"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</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">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</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">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/>
<b>hence </b><a NAME="E12:15_1"/>
 <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">x0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E2:15"/><i><font color="Green">E45</font></i>: 
<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>and </b><br/><a NAME="E3:15"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">x0</font>, <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> holds <br/> ex <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> &amp; ( for <font color="Olive">x1</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;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">x0</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</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">x0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> ) )
 ;<br/>

<i><font color="Green">E47</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<font color="Maroon">X</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x0</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="E1:15_3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:15_3"/>
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> holds <br/> ex <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> &amp; ( for <font color="Olive">x1</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>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> holds <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">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</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="E1:15_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/>

<b>then 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="E3:15_3_1"/><i><font color="Green">E51</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> &amp; ( for <font color="Olive">x1</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;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</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="Maroon">x0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> ) )
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">s</font>
;<br/>

<b>thus </b><a NAME="E4:15_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<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>assume </b><a NAME="E5:15_3_1"/><i><font color="Green">E53</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">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> )
 ;<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">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</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">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</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">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</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"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<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="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">x0</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/>
<b>hence </b><a NAME="E7:15_3_1"/>
 <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">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</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">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:15_3"/>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font> <a href="fcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">x0</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:15"/>
<font color="Maroon">f</font> <a href="fcont_1.html#R2" target="_self">is_continuous_on</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>


</div>
