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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>let </b><font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>







<b>assume </b><a NAME="E1:81"/>
<font color="Maroon">f</font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E3:81"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>  st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> holds <br/><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="partfun2.html#D3">PARTFUN2:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>;<br/><b>assume </b><a NAME="E1:81_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="comseq_1.html#K4">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:81_1"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="comseq_1.html#K4">(#)</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:81_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/><a NAME="E4:81_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="cfunct_1.html#T21" target="_self">Th21</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="comseq_1.html#K4">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font> = 
<font color="Maroon">q</font> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="cfunct_1.html#T21" target="_self">Th21</a>, </i>
<br/>.= 
<font color="Maroon">q</font> <a href="complex1.html#K9">*</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="cfunct_1.html#T20" target="_self">Th20</a>, <a class="ref" href="cfunct_1.html#T23" target="_self">Th23</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:81"/>
<font color="Maroon">q</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="partfun2.html#D3">PARTFUN2:def 3</a></i>;<br/>


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