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<b>let </b><font color="Maroon">X</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="E1:26"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">f1</font> <a href="fcont_1.html#R2" target="_self">is_continuous_on</a> <font color="Maroon">X</font> &amp; <font color="Maroon">f2</font> <a href="fcont_1.html#R2" target="_self">is_continuous_on</a> <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> )
 ;<br/>

<a NAME="E2:26"/><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">f1</font> &amp; <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</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/>
<b>hence </b><a NAME="E3:26"/>
<font color="Maroon">X</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">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_3.html#T3">FUNCT_3:3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="fcont_1.html#D2" target="_self">FCONT_1:def 2</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="E4:26"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E5:26"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">f1</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">r</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:26"/>
<font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E7:26"/><b>then </b>
<font color="Maroon">f2</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> <a href="fcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:26"/><b>then </b>
<font color="Maroon">f2</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> <a href="fcont_1.html#R1" target="_self">is_continuous_in</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E9:26"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="fcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">r</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E10:26"/>
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span> <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">r</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T2">RFUNCT_2:2</a></i>;<br/>


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