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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">Y</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:13"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">Y</font> is <a href="rcomp_1.html#V1">compact</a> &amp; <font color="Maroon">f</font> <a href="fcont_2.html#R1" target="_self">is_uniformly_continuous_on</a> <font color="Maroon">Y</font> )
 ;<br/>

<a NAME="E2:13"/><b>then </b>
<font color="Maroon">f</font> <a href="fcont_1.html#R2">is_continuous_on</a> <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:13"/><b>then </b>
 ex <font color="Olive">x1</font>, <font color="Olive">x2</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K4">upper_bound</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x2</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="fcont_1.html#T32">FCONT_1:32</a></i>;<br/>
<b>hence </b><a NAME="E4:13"/>
 ex <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K4">upper_bound</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x2</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font></span>)</span> )
 ;<br/>


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