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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K2">COMPLEX</a> ;<br/>







<div><i><font color="Green">E7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="cfuncdom.html#K2" target="_self">ComplexFuncMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E2:8_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="funcsdom.html#K4">.:</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="cfuncdom.html#E6"><i><font color="Green">Lemma4</font></i></a></i>;<br/><b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="funcsdom.html#K4">.:</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="cfuncdom.html#E1:8_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="cfuncdom.html#D2" target="_self">Def2</a></i>
<br/>.= 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cfuncdom.html#E2:8_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D5">BINOP_2:def 5</a></i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_2"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:8_2_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="funcsdom.html#K4">.:</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="cfuncdom.html#E6"><i><font color="Green">Lemma4</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K2" target="_self">ComplexFuncMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="funcsdom.html#K4">.:</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="cfuncdom.html#D2" target="_self">Def2</a></i>
<br/>.= 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cfuncdom.html#E1:8_2_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D5">BINOP_2:def 5</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="cfuncdom.html#E1:8_2"><i><font color="Green">E8</font></i></a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:8_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="cfuncdom.html#K2" target="_self">ComplexFuncMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:8"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="cfuncdom.html#K2" target="_self">ComplexFuncMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> iff for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="cfuncdom.html#E1:8"><i><font color="Green">E7</font></i></a></i>;<br/>


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