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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="xcmplx_0.html#V1">complex</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<a NAME="E1:9"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K2">COMPLEX</a> 
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D2">XCMPLX_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  or  not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1"><b>suppose </b></a><a NAME="E1:9_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/><div class="add"><a NAME="E2:9_1_1"/><b>then </b>
 <a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> = 
 <a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="complex1.html#E1:9_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="complex1.html#D2" target="_self">Def2</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_2"><b>suppose </b></a><a NAME="E1:9_1_2"/><i><font color="Green">E11</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="funct_2.html#NM1">Function</a> of 2, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:9_1_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:9_1_2"/><i><font color="Green">E13</font></i>: 
 <a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="ref" href="complex1.html#D3" target="_self">Def3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_2.html#NM1">Function</a> of 2, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E6:9_1_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E7:9_1_2"/><i><font color="Green">E15</font></i>: 
 <a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 0
 <b>by </b><i><a class="txt" href="complex1.html#E1:9_1_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="complex1.html#D2" target="_self">Def2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E9:9_1_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="complex1.html#E3:9_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="complex1.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E10:9_1_2"/><i><font color="Green">E17</font></i>: 
 <a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="complex1.html#E6:9_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="complex1.html#E7:9_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="complex1.html#E9:9_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/><a NAME="E11:9_1_2"/><i><font color="Green">E18</font></i>: 
 <a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="complex1.html#E3:9_1_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="complex1.html#E4:9_1_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="complex1.html#E9:9_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/><a NAME="E12:9_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> 2,<a href="numbers.html#K1">REAL</a> </span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> 2,<a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 0 </span>}</span> 
 <b>by </b><i><a class="txt" href="complex1.html#E1:9"><i><font color="Green">E10</font></i></a>, <a class="txt" href="complex1.html#E1:9_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="complex1.html#E1"><i><font color="Green">Lemma1</font></i></a>, <a class="txt" href="complex1.html#E2"><i><font color="Green">Lemma2</font></i></a>, <a class="ref" href="numbers.html#D2">NUMBERS:def 2</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E13:9_1_2"/><b>then </b><i><font color="Green">E19</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> 2,<a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 0 </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>1</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> 2,<a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="complex1.html#E9:9_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="complex1.html#E1"><i><font color="Green">Lemma1</font></i></a>, <a class="txt" href="complex1.html#E2"><i><font color="Green">Lemma2</font></i></a>, <a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/><a NAME="E15:9_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="complex1.html#E13:9_1_2"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E16:9_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="complex1.html#E9:9_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/><b>hence </b><a NAME="E17:9_1_2"/>
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="complex1.html#E9:9_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="complex1.html#E10:9_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="complex1.html#E11:9_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="complex1.html#E1"><i><font color="Green">Lemma1</font></i></a>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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