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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">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="E1:9"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">f</font> <a href="fcont_2.html#R1" target="_self">is_uniformly_continuous_on</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:9"/><b>then </b><i><font color="Green">E42</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">f</font> &amp; ( for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> holds <br/> ex <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> &amp; ( for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">x1</font> <a href="real_1.html#K5">-</a> <font color="Olive">x2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">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="Olive">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> ) ) ) )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><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> , <font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><a NAME="E1:9_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">x0</font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">r</font> is   <a href="real_1.html#NM1">Real</a> )
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>assume </b><a NAME="E2:9_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 ;<br/><b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:9_1"/><i><font color="Green">E45</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> &amp; ( for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">x1</font> <a href="real_1.html#K5">-</a> <font color="Olive">x2</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="Olive">x2</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">s</font> = <font color="Maroon">s</font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/><b>take </b><font color="Maroon">s</font> = <font color="Maroon">s</font>;<br/><b>thus </b><a NAME="E6:9_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<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/><a NAME="E7:9_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x1</font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>assume </b><a NAME="E8:9_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</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/><b>hence </b><a NAME="E9:9_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>, , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:9"/>
<font color="Maroon">f</font> <a href="fcont_1.html#R2">is_continuous_on</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="fcont_1.html#T15">FCONT_1:15</a></i>;<br/>


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