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<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">C</font> be  non <a href="xboole_0.html#V1">empty</a>  <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 <font color="Maroon">C</font>, <a href="numbers.html#K2">COMPLEX</a> ;<br/>





<b>assume </b><a NAME="E1:84"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">Y</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><a NAME="E1:84_1"/>
<font color="Maroon">r</font> <a href="comseq_1.html#K4">(#)</a> <font color="Maroon">f</font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="cfunct_1.html#T23" target="_self">Th23</a>, <a class="ref" href="cfunct_1.html#T19" target="_self">Th19</a></i>;<br/><b>hence </b><a NAME="E2:84_1"/>
<font color="Maroon">r</font> <a href="comseq_1.html#K4">(#)</a> <font color="Maroon">f</font> <a href="cfunct_1.html#R1" target="_self">is_bounded_on</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="cfunct_1.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:84"/>
for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  holds  <font color="Olive">r</font> <a href="comseq_1.html#K4">(#)</a> <font color="Maroon">f</font> <a href="cfunct_1.html#R1" target="_self">is_bounded_on</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E4:84"/>
 <a href="comseq_1.html#K5">-</a> <font color="Maroon">f</font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="cfunct_1.html#T23" target="_self">Th23</a>, <a class="ref" href="cfunct_1.html#T20" target="_self">Th20</a></i>;<br/>
<b>hence </b><a NAME="E5:84"/>
 <a href="comseq_1.html#K5">-</a> <font color="Maroon">f</font> <a href="cfunct_1.html#R1" target="_self">is_bounded_on</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="cfunct_1.html#T21" target="_self">Th21</a></i>;<br/>

<a NAME="E6:84"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">f</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="cfunct_1.html#T23" target="_self">Th23</a>, <a class="ref" href="cfunct_1.html#T20" target="_self">Th20</a></i>;<br/>
<b>hence </b><a NAME="E7:84"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">f</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T108">RFUNCT_1:108</a></i>;<br/>


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