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<div class="add">

<span class="kw">thus </span><a NAME="E1:10_2_1"/>
( <font color="Maroon" title="c2">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  ex <font color="Olive" title="b1">z9</font> being   <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
  <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="c2">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  ex <font color="Olive">b<sub>1</sub></font> being   <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c5">y1</font> =  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <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="c3">x</font>,<font color="Maroon" title="c3">x</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="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">y2</font> =  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <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="c3">x</font>,<font color="Maroon" title="c3">x</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="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c7">z9</font> = <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#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">z9</font> = <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#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> as   <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D2" target="_self" title="XCMPLX_0:def.2">Def2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:10_2_1_1"/><span class="lab"><font color="Green" title="E7">A15</font></span>: 
<font color="Maroon" title="c2">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z9</font> being   <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c8">z9</font>
; <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="c2">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<span class="lab"><font color="Green" title="E8">A16</font></span>:  <a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>)</span>)</span> = 
 <a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<span class="p4">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p0">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p0">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T15" title="ARYTM_0:th.15">ARYTM_0:15</a></span>
<br/>.= 
 <a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <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="c4">y</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<span class="p4">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p0">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p0">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T15" title="ARYTM_0:th.15">ARYTM_0:15</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <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="c3">x</font>,<font color="Maroon" title="c3">x</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="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T13" title="ARYTM_0:th.13">ARYTM_0:13</a></span>
;<br/>
<a NAME="E4:10_2_1_1"/><span class="lab"><font color="Green" title="E9">A17</font></span>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <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="c3">x</font>,<font color="Maroon" title="c3">x</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="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T13" title="ARYTM_0:th.13">ARYTM_0:13</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E10">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:10_2_1_1_2"/>
 <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="c3">x</font>,<font color="Maroon" title="c3">x</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="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>) <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"> contradiction</span><br/><a NAME="E2:10_2_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">y</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="ref" href="arytm_0.html#T17" title="ARYTM_0:th.17">ARYTM_0:17</a></span>;<br/><span class="kw">hence </span><a NAME="E3:10_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E2:10_2_1_1"><span class="lab"><font color="Green" title="E7">A15</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/></div><span class="kw">end;</span></div>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>) = 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</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="c3">x</font>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T13" title="ARYTM_0:th.13">ARYTM_0:13</a></span>
<br/>.= 
 <a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T15" title="ARYTM_0:th.15">ARYTM_0:15</a></span>
;<br/>
<a NAME="E7:10_2_1_1"/><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="c3">x</font>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</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="ref" href="arytm_0.html#D3" title="ARYTM_0:def.3">ARYTM_0:def 3</a></span>;<br/>
<span class="kw">then </span><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="c3">x</font>,<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p5">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p0">(<span class="default"><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="c3">x</font>,<font color="Maroon" title="c3">x</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="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>)</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="c4">y</font>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p0">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p0">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>)</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="c3">x</font>,<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p0">(<span class="default"><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="c3">x</font>,<font color="Maroon" title="c3">x</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="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>)</span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p5">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p0">(<span class="default"><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="c3">x</font>,<font color="Maroon" title="c3">x</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="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>)</span>)</span>)</span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></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="c3">x</font>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>)</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="c4">y</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p0">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p0">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#D5" title="ARYTM_0:def.5">ARYTM_0:def 5</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <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="c3">x</font>,<font color="Maroon" title="c3">x</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="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_2_1_1"><span class="lab"><font color="Green" title="E8">A16</font></span></a>, <a class="txt" href="#E4:10_2_1_1"><span class="lab"><font color="Green" title="E9">A17</font></span></a>, <a class="ref" href="arytm_0.html#T14" title="ARYTM_0:th.14">ARYTM_0:14</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_2_1_1"><span class="lab"><font color="Green" title="E10">A18</font></span></a>, <a class="ref" href="arytm_0.html#D4" title="ARYTM_0:def.4">ARYTM_0:def 4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:10_2_1_1"/>
<font color="Maroon" title="c2">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">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="#E3:10"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="xcmplx_0.html#D5" target="_self" title="XCMPLX_0:def.5">Def5</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/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:10_2_1"/>
<font color="Maroon" title="c2">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">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Olive">b<sub>1</sub></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">hence </span><a NAME="E3:10_2_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Olive">b<sub>1</sub></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"> verum</span><br/>


</div>
