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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>assume </b><a NAME="E1:49"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">x0</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>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x0</font> <a href="hidden.html#R1">=</a> <font color="Olive">x0</font> <a href="square_1.html#K3">^2</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">f1</font> =  <a href="partfun2.html#K1">id</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E3:49"/><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/>
<a NAME="E4:49"/><b>then </b>
for <font color="Olive">x</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> holds <br/><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<a NAME="E5:49"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">f1</font> <a href="fcont_1.html#R2" target="_self">is_continuous_on</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:49"/><i><font color="Green">E42</font></i>: 
<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">f1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x0</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:49_1"/><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>
 ;<br/><b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x0</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><b>thus </b><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font> = 
<font color="Maroon">x1</font> <a href="square_1.html#K5">^2</a> 
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<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#K4">*</a> <font color="Maroon">x1</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E8:49"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">f1</font>
 
<b>by </b><i>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:49"/>
<font color="Maroon">f</font> <a href="fcont_1.html#R2" target="_self">is_continuous_on</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, </i>;<br/>


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