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<b>let </b><font color="Maroon" title="c5">z</font>, <font color="Maroon" title="c6">z'</font> be  <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:9_2_3"/><i><font color="Green" title="E7">A54</font></i>: 
( <font color="Maroon" title="c6">z'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 implies <font color="Maroon" title="c6">z'</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>and </b><br/><a NAME="E2:9_2_3"/><i><font color="Green" title="E8">A55</font></i>: 
( <font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 implies <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/>

<b>thus </b><a NAME="E3:9_2_3"/>
( <font color="Maroon" title="c5">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 implies <font color="Maroon" title="c5">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="xcmplx_0.html#E1:9_2_3"><i><font color="Green" title="E7">A54</font></i></a>, <a class="txt" href="xcmplx_0.html#E2:9_2_3"><i><font color="Green" title="E8">A55</font></i></a></i>;<br/>

<b>assume </b><a NAME="E4:9_2_3"/><i><font color="Green" title="E9">A56</font></i>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<b>assume </b><a NAME="E5:9_2_3"/>
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c8">x2</font>, <font color="Maroon" title="c9">y1</font>, <font color="Maroon" title="c10">y2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E7:9_2_3"/><i><font color="Green" title="E10">A57</font></i>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>and </b><br/><a NAME="E8:9_2_3"/>
<font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c9">y1</font>,<font color="Maroon" title="c10">y2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>and </b><br/><a NAME="E9:9_2_3"/><i><font color="Green" title="E11">A58</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c9">y1</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c10">y2</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c10">y2</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">y1</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>by </b><i><a class="txt" href="xcmplx_0.html#E1:9_2_3"><i><font color="Green" title="E7">A54</font></i></a>, <a class="ref" href="xcmplx_0.html#D5" target="_self" title="XCMPLX_0:def.5">Def5</a></i>;<br/>
<a NAME="E10:9_2_3"/>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default">0,0</span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 
<b>by </b><i><a class="txt" href="xcmplx_0.html#E4:9_2_3"><i><font color="Green" title="E9">A56</font></i></a>, <a class="ref" href="arytm_0.html#D7" title="ARYTM_0:def.7">ARYTM_0:def 7</a></i>;<br/>
<a NAME="E11:9_2_3"/><b>then </b><i><font color="Green" title="E12">A59</font></i>: 
( <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="xcmplx_0.html#E7:9_2_3"><i><font color="Green" title="E10">A57</font></i></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></i>;<br/>
<a NAME="E12:9_2_3"/><b>then </b><i><font color="Green" title="E13">A60</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="ref" href="arytm_0.html#T14" title="ARYTM_0:th.14">ARYTM_0:14</a></i>;<br/>
<a NAME="E13:9_2_3"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c10">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="xcmplx_0.html#E11:9_2_3"><i><font color="Green" title="E12">A59</font></i></a>, <a class="ref" href="arytm_0.html#T14" title="ARYTM_0:th.14">ARYTM_0:14</a></i>;<br/>
<a NAME="E14:9_2_3"/><b>then </b><i><font color="Green" title="E14">A61</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c9">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c10">y2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="xcmplx_0.html#E12:9_2_3"><i><font color="Green" title="E13">A60</font></i></a>, <a class="ref" href="arytm_0.html#D4" title="ARYTM_0:def.4">ARYTM_0:def 4</a></i>;<br/>
<a NAME="E15:9_2_3"/>
(  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c10">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="xcmplx_0.html#E11:9_2_3"><i><font color="Green" title="E12">A59</font></i></a>, <a class="ref" href="arytm_0.html#T14" title="ARYTM_0:th.14">ARYTM_0:14</a></i>;<br/>
<a NAME="E16:9_2_3"/><b>then </b>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c10">y2</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">y1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="ref" href="arytm_0.html#T13" title="ARYTM_0:th.13">ARYTM_0:13</a></i>;<br/>
<b>hence </b><a NAME="E17:9_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="xcmplx_0.html#E9:9_2_3"><i><font color="Green" title="E11">A58</font></i></a>, <a class="txt" href="xcmplx_0.html#E14:9_2_3"><i><font color="Green" title="E14">A61</font></i></a>, <a class="ref" href="arytm_0.html#D7" title="ARYTM_0:def.7">ARYTM_0:def 7</a></i>;<br/>


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