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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</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>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>3</sub></font>,<a href="numbers.html#K2">COMPLEX</a> ;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:25"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>and </b><br/><a NAME="E2:25"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K6">1r</a>  ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>and </b><br/><a NAME="E3:25"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K6">1r</a>  ) )
 ;<br/>

<a NAME="E4:25"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K6">1r</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K6">1r</a>  )
 
<b>by </b><i><a class="txt" href="cfuncdom.html#E1:25"><i><font color="Green">E25</font></i></a>, <a class="txt" href="cfuncdom.html#E2:25"><i><font color="Green">E26</font></i></a>, <a class="txt" href="cfuncdom.html#E3:25"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="cfuncdom.html#E1:25"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/>



<b>assume </b><a NAME="E6:25"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">c<sub>3</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>3</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>8</sub></font>,<font color="Maroon">c<sub>4</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>3</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> <a href="funct_2.html#R2">=</a>  <a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then </b><i><font color="Green">E30</font></i>:  <a href="complex1.html#K5">0c</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>3</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <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>3</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>8</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<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>3</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>9</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="cfuncdom.html#T4" target="_self">Th4</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" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>3</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>8</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><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>3</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>9</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="cfuncdom.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><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>3</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>9</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="cfuncdom.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="complex1.html#K9">*</a> <a href="complex1.html#K6">1r</a> </span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="complex1.html#K9">*</a> <a href="complex1.html#K5">0c</a> </span>)</span>
<b>by </b><i><a class="txt" href="cfuncdom.html#E4:25"><i><font color="Green">E28</font></i></a>, <a class="ref" href="cfuncdom.html#T6" target="_self">Th6</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="complex1.html#K9">*</a> <a href="complex1.html#K5">0c</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="complex1.html#K8">+</a> <a href="complex1.html#K5">0c</a> 
<b>by </b><i><a class="ref" href="complex1.html#D6">COMPLEX1:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="complex1.html#D6">COMPLEX1:def 6</a></i>
;<br/>
 <a href="complex1.html#K5">0c</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>3</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <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>3</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>8</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<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>3</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>9</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="cfuncdom.html#E6:25"><i><font color="Green">E29</font></i></a>, <a class="ref" href="cfuncdom.html#T4" target="_self">Th4</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" target="_self">ComplexFuncExtMult</a> <font color="Maroon">c<sub>3</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>8</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><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>3</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>9</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="cfuncdom.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><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>3</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>9</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="cfuncdom.html#T6" target="_self">Th6</a></i>
<br/>.= 
<a href="complex1.html#K5">0c</a>  <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><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>3</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>9</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cfuncdom.html#E4:25"><i><font color="Green">E28</font></i></a>, <a class="ref" href="complex1.html#D6">COMPLEX1:def 6</a></i>
<br/>.= 
<a href="complex1.html#K5">0c</a>  <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="complex1.html#K9">*</a> <a href="complex1.html#K6">1r</a> </span>)</span>
<b>by </b><i><a class="txt" href="cfuncdom.html#E4:25"><i><font color="Green">E28</font></i></a>, <a class="ref" href="cfuncdom.html#T6" target="_self">Th6</a></i>
<br/>.= 
<a href="complex1.html#K5">0c</a>  <a href="complex1.html#K8">+</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="complex1.html#D6">COMPLEX1:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E9:25"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a>  &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a>  )
 <b>by </b><i><a class="txt" href="cfuncdom.html#E7:25"><i><font color="Green">E30</font></i></a></i>;<br/>


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