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<span class="kw">let </span><font color="Maroon" title="c5">z</font>, <font color="Maroon" title="c6">z9</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c6">z9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c6">z9</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 ) &amp; (  not <font color="Maroon" title="c6">z9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) implies ( ( <font color="Maroon" title="c5">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; (  not <font color="Maroon" title="c5">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c6">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:10_2_3"/><span class="lab"><font color="Green" title="E7">A54</font></span>: 
( <font color="Maroon" title="c6">z9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c6">z9</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 )
 <span class="kw">and </span><br/><a NAME="E2:10_2_3"/><span class="lab"><font color="Green" title="E8">A55</font></span>: 
( <font color="Maroon" title="c6">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c5">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; (  not <font color="Maroon" title="c5">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c6">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E3:10_2_3"/>
( <font color="Maroon" title="c5">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2_3"><span class="lab"><font color="Green" title="E7">A54</font></span></a>, <a class="txt" href="#E2:10_2_3"><span class="lab"><font color="Green" title="E8">A55</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c6">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E4:10_2_3"/><span class="lab"><font color="Green" title="E9">A56</font></span>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<span class="kw">assume </span><a NAME="E5:10_2_3"/>
<font color="Maroon" title="c6">z9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E7:10_2_3"/><span class="lab"><font color="Green" title="E10">A57</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E8:10_2_3"/>
<font color="Maroon" title="c6">z9</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>
 <span class="kw">and </span><br/><a NAME="E9:10_2_3"/><span class="lab"><font color="Green" title="E11">A58</font></span>: 
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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2_3"><span class="lab"><font color="Green" title="E7">A54</font></span></a>, <a class="ref" href="xcmplx_0.html#D5" target="_self" title="XCMPLX_0:def.5">Def5</a></span>;<br/>
<a NAME="E10:10_2_3"/><span class="lab"><font color="Green" title="E12">A59</font></span>: 
<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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_2_3"><span class="lab"><font color="Green" title="E9">A56</font></span></a>, <a class="ref" href="arytm_0.html#D5" title="ARYTM_0:def.5">ARYTM_0:def 5</a></span>;<br/>
<a NAME="E11:10_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A60</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10_2_3"><span class="lab"><font color="Green" title="E10">A57</font></span></a>, <a class="ref" href="arytm_0.html#T10" title="ARYTM_0:th.10">ARYTM_0:10</a></span>;<br/>
<a NAME="E12:10_2_3"/><span class="lab"><font color="Green" title="E14">A61</font></span>: 
<font color="Maroon" title="c8">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10_2_3"><span class="lab"><font color="Green" title="E10">A57</font></span></a>, <a class="txt" href="#E10:10_2_3"><span class="lab"><font color="Green" title="E12">A59</font></span></a>, <a class="ref" href="arytm_0.html#T10" title="ARYTM_0:th.10">ARYTM_0:10</a></span>;<br/>
<a NAME="E13:10_2_3"/><span class="lab"><font color="Green" title="E15">A62</font></span>: 
 <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:10_2_3"><span class="lab"><font color="Green" title="E13">A60</font></span></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></span>;<br/>
<a NAME="E14:10_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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:10_2_3"><span class="lab"><font color="Green" title="E14">A61</font></span></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></span>;<br/>
<a NAME="E15:10_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A63</font></span>: 
 <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:10_2_3"><span class="lab"><font color="Green" title="E15">A62</font></span></a>, <a class="ref" href="arytm_0.html#D3" title="ARYTM_0:def.3">ARYTM_0:def 3</a></span>;<br/>
<a NAME="E16:10_2_3"/><span class="lab"><font color="Green" title="E17">A64</font></span>: 
 <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:10_2_3"><span class="lab"><font color="Green" title="E13">A60</font></span></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></span>;<br/>
<a NAME="E17:10_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="c9">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:10_2_3"><span class="lab"><font color="Green" title="E14">A61</font></span></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></span>;<br/>
<a NAME="E18:10_2_3"/><span class="kw">then </span>
 <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:10_2_3"><span class="lab"><font color="Green" title="E17">A64</font></span></a>, <a class="ref" href="arytm_0.html#T11" title="ARYTM_0:th.11">ARYTM_0:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:10_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:10_2_3"><span class="lab"><font color="Green" title="E11">A58</font></span></a>, <a class="txt" href="#E15:10_2_3"><span class="lab"><font color="Green" title="E16">A63</font></span></a>, <a class="ref" href="arytm_0.html#D5" title="ARYTM_0:def.5">ARYTM_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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