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<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="E1:5"/>
( <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 <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">N</font> <a href="tarski.html#R1">c=</a> <font color="Olive">N1</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_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="E2:5_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">N</font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E4:5_1"/><i><font color="Green">E39</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">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="Maroon">N1</font>
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">N</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><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="E1:5_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N</font>
 ;<br/><a NAME="E2:5_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Olive">x</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">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T78">PARTFUN2:78</a></i>;<br/><b>hence </b><a NAME="E3:5_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:5_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N1</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T1">RCOMP_1:1</a></i>;<br/>


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

<b>assume </b><a NAME="E2:5"/><i><font color="Green">E41</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 <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">N</font> <a href="tarski.html#R1">c=</a> <font color="Olive">N1</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>now </b></a><div class="add"><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">N</font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E2:5_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N1</font>
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">N</font> = <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="E3:5_2"/>
( <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="E4:5_2"/><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">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>hence </b><a NAME="E5:5_2"/>
<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/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:5"/>
<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/>


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