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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<a href="topmetr.html#K3">R^1</a> ;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>





<b>assume </b><a NAME="E1:23"/>
( <font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">P</font> is <a href="compts_1.html#V6">compact</a> &amp; <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a> )
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>, <font color="Maroon">x2</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:23"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K4">upper_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3" target="_self">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span></span>)</span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3" target="_self">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">x1</font>
;<br/>

<b>thus </b><a NAME="E4:23"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="weierstr.html#K4" target="_self">upper_bound</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span> )
 <b>by </b><i/>;<br/>


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