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

<b>consider </b><font color="Maroon">c<sub>1</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:33"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="cfuncdom.html#E34"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>4</sub></font> =  <a href="cfuncdom.html#K6" target="_self">ComplexVectSpace</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being    <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> <b> such that </b><br/><a NAME="E4:33"/><i><font color="Green">E30</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cfunct_1.html#NM1">Complex</a> st <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="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</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="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a>  ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <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> ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="cfunct_1.html#NM1">Complex</a> st <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#R2">=</a> <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="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>5</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="Olive">b<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="cfuncdom.html#E2:33"><i><font color="Green">E29</font></i></a>, <a class="ref" href="cfuncdom.html#T24" target="_self">Th24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>4</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E6:33"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cfunct_1.html#NM1">Complex</a> st <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/>

<b>assume </b><a NAME="E1:33_1"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>8</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/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</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#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="cfuncdom.html#E1:33_1"><i><font color="Green">E31</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="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>9</sub></font>,<font color="Maroon">c<sub>5</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>10</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>

;<br/>
<a NAME="E4:33_1"/><b>then </b>
<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>9</sub></font>,<font color="Maroon">c<sub>5</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>10</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<b>hence </b><a NAME="E5:33_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="cfuncdom.html#E4:33"><i><font color="Green">E30</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E7:33"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>4</sub></font>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="cfunct_1.html#NM1">Complex</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</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>consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <a href="cfunct_1.html#NM1">Complex</a><b> such that </b><br/><a NAME="E3:33_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#R2">=</a> <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>11</sub></font>,<font color="Maroon">c<sub>5</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>12</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="cfuncdom.html#E4:33"><i><font color="Green">E30</font></i></a></i>;<br/>
<font color="Maroon">c<sub>10</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="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>7</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>12</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="cfuncdom.html#E3:33_2"><i><font color="Green">E31</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>11</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>8</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="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>

;<br/>
<b>hence </b><a NAME="E5:33_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cfunct_1.html#NM1">Complex</a> st <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/>


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


</div>
