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<b>let </b><font color="Maroon" title="c1">x1</font>, <font color="Maroon" title="c2">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">A</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="c4">f</font>, <font color="Maroon" title="c5">g</font> be    <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="c3">A</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:25"/><i><font color="Green" title="E23">A1</font></i>: 
( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x2</font> )
 <b>and </b><br/><a NAME="E2:25"/><i><font color="Green" title="E24">A2</font></i>: 
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> holds <br/>( ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> implies <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  ) &amp; ( <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x1</font> implies <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 <b>and </b><br/><a NAME="E3:25"/><i><font color="Green" title="E25">A3</font></i>: 
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> holds <br/>( ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> implies <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x1</font> implies <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  ) )
 ;<br/>

<a NAME="E4:25"/><i><font color="Green" title="E26">A4</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  &amp; <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  &amp; <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  )
 
<b>by </b><i><a class="txt" href="cfuncdom.html#E1:25"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="cfuncdom.html#E2:25"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="cfuncdom.html#E3:25"><i><font color="Green" title="E25">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c1">x1</font>, <font color="Maroon" title="c7">x2</font> = <font color="Maroon" title="c2">x2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> <b>by </b><i><a class="txt" href="cfuncdom.html#E1:25"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>
<b>let </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>;<br/>



<b>assume </b><a NAME="E6:25"/><i><font color="Green" title="E27">A5</font></i>: 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" title="CFUNCDOM:func.1">ComplexFuncAdd</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funcsdom.html#K1" title="FUNCSDOM:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K3" title="CFUNCDOM:func.3">ComplexFuncExtMult</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funcsdom.html#K2" title="FUNCSDOM:func.2">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c4">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K3" title="CFUNCDOM:func.3">ComplexFuncExtMult</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funcsdom.html#K2" title="FUNCSDOM:func.2">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c5">g</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cfuncdom.html#K4" title="CFUNCDOM:func.4">ComplexFuncZero</a> <font color="Maroon" title="c3">A</font>
 ;<br/>

<b>then </b><i><font color="Green" title="E28">A6</font></i>:  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K1" title="CFUNCDOM:func.1">ComplexFuncAdd</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funcsdom.html#K1" title="FUNCSDOM:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cfuncdom.html#K3" title="CFUNCDOM:func.3">ComplexFuncExtMult</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funcsdom.html#K2" title="FUNCSDOM:func.2">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c4">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cfuncdom.html#K3" title="CFUNCDOM:func.3">ComplexFuncExtMult</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funcsdom.html#K2" title="FUNCSDOM:func.2">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c5">g</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x1</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cfuncdom.html#K3" title="CFUNCDOM:func.3">ComplexFuncExtMult</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funcsdom.html#K2" title="FUNCSDOM:func.2">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c4">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cfuncdom.html#K3" title="CFUNCDOM:func.3">ComplexFuncExtMult</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funcsdom.html#K2" title="FUNCSDOM:func.2">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c5">g</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x1</font></span>)</span>
<b>by </b><i><a class="ref" href="cfuncdom.html#T1" target="_self" title="CFUNCDOM:th.1">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x1</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cfuncdom.html#K3" title="CFUNCDOM:func.3">ComplexFuncExtMult</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funcsdom.html#K2" title="FUNCSDOM:func.2">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c5">g</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x1</font></span>)</span>
<b>by </b><i><a class="ref" href="cfuncdom.html#T6" target="_self" title="CFUNCDOM:th.6">Th6</a></i>
<br/>.= 
<font color="Maroon" title="c8">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> </span>)</span>
<b>by </b><i><a class="txt" href="cfuncdom.html#E4:25"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="cfuncdom.html#T6" target="_self" title="CFUNCDOM:th.6">Th6</a>, <a class="ref" href="complex1.html#D7" title="COMPLEX1:def.7">COMPLEX1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c8">a</font>

;<br/>
 <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K1" title="CFUNCDOM:func.1">ComplexFuncAdd</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funcsdom.html#K1" title="FUNCSDOM:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cfuncdom.html#K3" title="CFUNCDOM:func.3">ComplexFuncExtMult</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funcsdom.html#K2" title="FUNCSDOM:func.2">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c4">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cfuncdom.html#K3" title="CFUNCDOM:func.3">ComplexFuncExtMult</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funcsdom.html#K2" title="FUNCSDOM:func.2">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c5">g</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x2</font>
<b>by </b><i><a class="txt" href="cfuncdom.html#E6:25"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cfuncdom.html#K3" title="CFUNCDOM:func.3">ComplexFuncExtMult</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funcsdom.html#K2" title="FUNCSDOM:func.2">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c4">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cfuncdom.html#K3" title="CFUNCDOM:func.3">ComplexFuncExtMult</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funcsdom.html#K2" title="FUNCSDOM:func.2">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c5">g</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x2</font></span>)</span>
<b>by </b><i><a class="ref" href="cfuncdom.html#T1" target="_self" title="CFUNCDOM:th.1">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cfuncdom.html#K3" title="CFUNCDOM:func.3">ComplexFuncExtMult</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funcsdom.html#K2" title="FUNCSDOM:func.2">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c5">g</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x2</font></span>)</span>
<b>by </b><i><a class="ref" href="cfuncdom.html#T6" target="_self" title="CFUNCDOM:th.6">Th6</a></i>
<br/>.= 
<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span>
<b>by </b><i><a class="txt" href="cfuncdom.html#E4:25"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="cfuncdom.html#T6" target="_self" title="CFUNCDOM:th.6">Th6</a></i>
<br/>.= 
<font color="Maroon" title="c9">b</font>
<b>by </b><i><a class="ref" href="complex1.html#D7" title="COMPLEX1:def.7">COMPLEX1:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E9:25"/>
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  )
 <b>by </b><i><a class="txt" href="cfuncdom.html#E7:25"><i><font color="Green" title="E28">A6</font></i></a></i>;<br/>


</div>
