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<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></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#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c4">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c5">y</font></span>)</span></span><br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1"><b>suppose </b></a><a NAME="E1:27_1_1"/><i><font color="Green" title="E25">A1</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">y</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></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#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c4">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c5">y</font></span>)</span></span><br/><div class="add"><a NAME="E2:27_1_1"/><b>then </b><i><font color="Green" title="E26">A2</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="arytm_0.html#T14" target="_self" title="ARYTM_0:th.14">Th14</a></i>;<br/><a NAME="E3:27_1_1"/><i><font color="Green" title="E27">A3</font></i>: 
(  <a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:27_1_1"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="arytm_0.html#D5" target="_self" title="ARYTM_0:def.5">Def5</a></i>;<br/><b>thus </b> <a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="arytm_0.html#E2:27_1_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="arytm_0.html#D5" target="_self" title="ARYTM_0:def.5">Def5</a></i>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c4">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c5">y</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E3:27_1_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="arytm_0.html#T14" target="_self" title="ARYTM_0:th.14">Th14</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:27_1_2"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E2:27_1_2"/><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c5">y</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></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#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c4">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c5">y</font></span>)</span></span><br/><div class="add"><a NAME="E3:27_1_2"/><i><font color="Green" title="E27">A6</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">y</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="arytm_0.html#E2:27_1_2"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="arytm_0.html#D5" target="_self" title="ARYTM_0:def.5">Def5</a></i>;<br/><a NAME="E4:27_1_2"/><i><font color="Green" title="E28">A7</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:27_1_2"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="arytm_0.html#D5" target="_self" title="ARYTM_0:def.5">Def5</a></i>;<br/><a NAME="E5:27_1_2"/><i><font color="Green" title="E29">A8</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:27_1_2"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="arytm_0.html#E2:27_1_2"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="arytm_0.html#T23" target="_self" title="ARYTM_0:th.23">Th23</a></i>;<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">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<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#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c4">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c5">y</font></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#K2" title="ARYTM_0:func.2">*</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c5">y</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T15" target="_self" title="ARYTM_0:th.15">Th15</a></i>
<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="c3">j</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c5">y</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E4:27_1_2"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="arytm_0.html#T15" target="_self" title="ARYTM_0:th.15">Th15</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="arytm_0.html#E3:27_1_2"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="arytm_0.html#T21" target="_self" title="ARYTM_0:th.21">Th21</a></i>
;<br/><b>hence </b><a NAME="E7:27_1_2"/>
 <a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></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#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c4">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c5">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="arytm_0.html#E5:27_1_2"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="arytm_0.html#D5" target="_self" title="ARYTM_0:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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