<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">Y</font> be   <a href="clvect_1.html#NM2" title="CLVECT_1:NM.2">ComplexLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">h</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K3" title="CLOPBAN1:func.3">ComplexVectSpace</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c5">c</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>;<br/>







<b>reconsider </b><font color="Maroon" title="c6">f'</font> = <font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c7">h'</font> = <font color="Maroon" title="c4">h</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> ;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:20_1"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">f</font>
 ;<br/><b>let </b><font color="Maroon" title="c8">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><a NAME="E2:20_1"/>
<font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban1.html#K2" title="CLOPBAN1:func.2">FuncExtMult</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="funcsdom.html#K2" title="FUNCSDOM:func.2">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">f'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="clopban1.html#E1:20_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="clvect_1.html#D1" title="CLVECT_1:def.1">CLVECT_1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:20_1"/>
<font color="Maroon" title="c4">h</font> <a href="clopban1.html#K4" title="CLOPBAN1:func.4">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="clopban1.html#K4" title="CLOPBAN1:func.4">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="clopban1.html#T3" target="_self" title="CLOPBAN1:th.3">Th3</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:20"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c4">h</font> <a href="clopban1.html#K4" title="CLOPBAN1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="clopban1.html#K4" title="CLOPBAN1:func.4">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 ;<br/>

<a NAME="E4:20"/><b>then </b>
<font color="Maroon" title="c7">h'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="clopban1.html#K2" title="CLOPBAN1:func.2">FuncExtMult</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="funcsdom.html#K2" title="FUNCSDOM:func.2">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">f'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="ref" href="clopban1.html#T3" target="_self" title="CLOPBAN1:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E5:20"/>
<font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="ref" href="clvect_1.html#D1" title="CLVECT_1:def.1">CLVECT_1:def 1</a></i>;<br/>


</div>
