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<b>let </b><font color="Maroon">c<sub>1</sub></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><b>that </b><br/><a NAME="E1:40"/><i><font color="Green">E18</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V3">bounded</a>
 <b>and </b><br/><a NAME="E2:40"/><i><font color="Green">E19</font></i>: 
 <a href="seq_4.html#K4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:40"/><i><font color="Green">E20</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> <a href="partfun2.html#R1">is_constant_on</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E5:40"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="rfunct_2.html#E3:40"><i><font color="Green">E20</font></i></a>, <a class="ref" href="partfun2.html#T77">PARTFUN2:77</a></i>;<br/>
<a NAME="E6:40"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E5:40"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:40"/>
contradiction
 <b>by </b><i><a class="txt" href="rfunct_2.html#E1:40"><i><font color="Green">E18</font></i></a>, <a class="txt" href="rfunct_2.html#E2:40"><i><font color="Green">E19</font></i></a>, <a class="txt" href="rfunct_2.html#E5:40"><i><font color="Green">E21</font></i></a>, <a class="ref" href="seq_4.html#T25">SEQ_4:25</a></i>;<br/>


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