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

<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>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>thus </b><a NAME="E1:4"/>
( <font color="Maroon">f</font> <a href="fcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">x0</font> implies ( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">N1</font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font>  ex <font color="Olive">N</font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font> st <br/>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> <font color="Maroon">f</font> &amp; <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Olive">N</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Olive">N1</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> <a href="fcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">x0</font>
 ;<br/>

<b>hence </b><a NAME="E2:4_1"/>
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">N1</font> be    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font>;<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="E4:4_1"/><i><font color="Green">E39</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">N1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i><a class="ref" href="rcomp_1.html#D7">RCOMP_1:def 7</a></i>;<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="E6:4_1"/><i><font color="Green">E41</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> <font color="Maroon">f</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>reconsider </b><font color="Maroon">N</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> as    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font> <b>by </b><i>, <a class="ref" href="rcomp_1.html#D7">RCOMP_1:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon">N</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>assume </b><a NAME="E8:4_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 ;<br/>

<a NAME="E9:4_1"/><b>then </b>
 <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>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/>
<a NAME="E10:4_1"/><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="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E11:4_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/>


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

<b>assume </b><a NAME="E2:4"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">N1</font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font>  ex <font color="Olive">N</font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font> st <br/>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> <font color="Maroon">f</font> &amp; <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Olive">N</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Olive">N1</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>now </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="E1:4_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/><b>then reconsider </b><font color="Maroon">N1</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> as    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font> <b>by </b><i><a class="ref" href="rcomp_1.html#D7">RCOMP_1:def 7</a></i>;<br/><b>consider </b><font color="Maroon">N2</font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E4:4_2"/><i><font color="Green">E44</font></i>: 
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> <font color="Maroon">f</font> &amp; <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N2</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font>
 <b>by </b><i/>;<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="E6:4_2"/><i><font color="Green">E45</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">N2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i><a class="ref" href="rcomp_1.html#D7">RCOMP_1:def 7</a></i>;<br/><b>take </b><font color="Maroon">s</font> = <font color="Maroon">s</font>;<br/><a NAME="E7:4_2"/>
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> <font color="Maroon">f</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1"><b>proof </b></a><div class="add">

<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="E1:4_2_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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/>

<a NAME="E2:4_2_1"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N2</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/>
<a NAME="E3:4_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:4_2_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><a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:4_2"/>
( 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> <font color="Maroon">f</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/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:4"/>
<font color="Maroon">f</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>
