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

<b>assume </b><b>that </b><br/><a NAME="E1:9_2_3"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">z'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies <font color="Maroon">z'</font> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 1 )
 <b>and </b><br/><a NAME="E2:9_2_3"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<b>thus </b><a NAME="E3:9_2_3"/>
( <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies <font color="Maroon">z</font> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, </i>;<br/>

<b>assume </b><a NAME="E4:9_2_3"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>assume </b><a NAME="E5:9_2_3"/>
<font color="Maroon">z'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>then 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="E7:9_2_3"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">z</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="E8:9_2_3"/>
<font color="Maroon">z'</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="E9:9_2_3"/><i><font color="Green">E101</font></i>: 
1 <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> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x1</font>,<font color="Maroon">y2</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x2</font>,<font color="Maroon">y1</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i>, </i>;<br/>
<a NAME="E10:9_2_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default">0,0</span><a href="arytm_0.html#K5">*]</a></span>
 
<b>by </b><i>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/>
<a NAME="E11:9_2_3"/><b>then </b><i><font color="Green">E102</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, <a class="ref" href="arytm_0.html#T12">ARYTM_0:12</a></i>;<br/>
<a NAME="E12:9_2_3"/><b>then </b><i><font color="Green">E103</font></i>: 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>;<br/>
<a NAME="E13:9_2_3"/>
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>;<br/>
<a NAME="E14:9_2_3"/><b>then </b><i><font color="Green">E104</font></i>: 
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="arytm_0.html#D4">ARYTM_0:def 4</a></i>;<br/>
<a NAME="E15:9_2_3"/>
(  <a href="arytm_0.html#K2">*</a> <font color="Maroon">x1</font>,<font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="arytm_0.html#K2">*</a> <font color="Maroon">x2</font>,<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, <a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>;<br/>
<a NAME="E16:9_2_3"/><b>then </b>
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x1</font>,<font color="Maroon">y2</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x2</font>,<font color="Maroon">y1</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>;<br/>
<b>hence </b><a NAME="E17:9_2_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/>


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