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

<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="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="cfuncdom.html#K8" target="_self">CAlgebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/>







<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="cfuncdom.html#K8" target="_self">CAlgebra</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <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/>
<b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="cfuncdom.html#K8" target="_self">CAlgebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="cfuncdom.html#T7" target="_self">Th7</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font>

;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="cfuncdom.html#K8" target="_self">CAlgebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <span class="p2">(<span class="default"><a href="cfuncdom.html#K8" target="_self">CAlgebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</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="ref" href="cfuncdom.html#T8" target="_self">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>

;<br/>

<b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="cfuncdom.html#K8" target="_self">CAlgebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="cfuncdom.html#K8" target="_self">CAlgebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="cfuncdom.html#T7" target="_self">Th7</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="cfuncdom.html#T12" target="_self">Th12</a></i>
;<br/>

<b>thus </b><a NAME="E5:48"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="cfuncdom.html#K8" target="_self">CAlgebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="cfuncdom.html#K8" target="_self">CAlgebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <a href="complex1.html#K6">1r</a> </span>)</span>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <a href="complex1.html#K6">1r</a> </span>)</span>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="cfuncdom.html#K8" target="_self">CAlgebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>12</sub></font> = 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <a href="complex1.html#K6">1r</a> </span>)</span>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>

<br/>.= 
 <a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="cfuncdom.html#T13" target="_self">Th13</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="cfuncdom.html#K8" target="_self">CAlgebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>

;<br/>


</div><b>end;</b></div>

<b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> = 
<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>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>

<br/>.= 
<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>9</sub></font>,<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="cfuncdom.html#T9" target="_self">Th9</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font>

;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> = 
<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="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>

<br/>.= 
<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> <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>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>

<br/>.= 
<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>8</sub></font>,<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>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="cfuncdom.html#T10" target="_self">Th10</a></i>
<br/>.= 
<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="binop_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>

;<br/>

<b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="cfuncdom.html#K8" target="_self">CAlgebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> = 
<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="binop_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="cfuncdom.html#K8" target="_self">CAlgebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>

<br/>.= 
<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>8</sub></font>,<span class="p1">(<span class="default"><a href="cfuncdom.html#K5" target="_self">ComplexFuncUnit</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cfuncdom.html#E45"><i><font color="Green">Lemma35</font></i></a></i>
<br/>.= 
<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> <span class="p1">(<span class="default"><a href="cfuncdom.html#K5" target="_self">ComplexFuncUnit</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="cfuncdom.html#T9" target="_self">Th9</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="cfuncdom.html#T11" target="_self">Th11</a></i>
;<br/>

<b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<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="binop_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>

<br/>.= 
<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>8</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <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>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span>,<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>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="cfuncdom.html#T17" target="_self">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<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>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>

;<br/>

<b>thus </b><font color="Maroon">c<sub>5</sub></font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<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>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="cfuncdom.html#T18" target="_self">Th18</a></i>
<br/>.= 
<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="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font>

;<br/>

<b>thus </b><font color="Maroon">c<sub>5</sub></font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="cfuncdom.html#E23"><i><font color="Green">Lemma20</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="complex1.html#K8">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="complex1.html#K8">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="cfuncdom.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="cfuncdom.html#T15" target="_self">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

;<br/>


</div>
