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<b>let </b><font color="Maroon">x</font> be  <a href="xcmplx_0.html#V1">complex</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">y</font> be  <a href="xcmplx_0.html#V1">complex</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">z</font> be  <a href="xcmplx_0.html#V1">complex</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:19"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font>
 ;<br/>

<b>consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y2</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="E3:19"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E4:19"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E5:19"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D4">XCMPLX_0:def 4</a></i>;<br/>
<a NAME="E6:19"/><i><font color="Green">E36</font></i>: 
(  <a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp;  <a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> )
 
<b>by </b><i><a class="txt" href="complex1.html#E3:19"><i><font color="Green">E22</font></i></a>, <a class="txt" href="complex1.html#E4"><i><font color="Green">Lemma23</font></i></a>, </i>;<br/>
<b>thus </b> <a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">z</font> = 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>
<b>by </b><i><a class="txt" href="complex1.html#E1:19"><i><font color="Green">E21</font></i></a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#D2" target="_self">Def2</a>, </i>
;<br/>


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