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<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">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> ;<br/>



<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1"/>
( <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>  or <font color="Maroon" title="c3">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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1"><b>suppose </b></a><a NAME="E1:24_1_1"/>
<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> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_1_1"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T14" target="_self" title="ARYTM_0:th.14">Th14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2"><b>suppose </b></a><a NAME="E1:24_1_2"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c3">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> 
 ;<br/><div class="add"><div><i><font color="Green" title="E26">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_2_1"/><i><font color="Green" title="E27">A4</font></i>: 
 <a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <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> 
 ;<br/><b>thus </b>1 = 
 <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> <font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_0.html#E1:24_1_2"><i><font color="Green" title="E25">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="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="arytm_0.html#E1:24_1_2_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="arytm_0.html#T14" target="_self" title="ARYTM_0:th.14">Th14</a></i>
;<br/></div><b>end;</b></div><a NAME="E3:24_1_2"/><i><font color="Green" title="E27">A5</font></i>: 
 <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> <font color="Maroon" title="c3">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#E1:24_1_2"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="arytm_0.html#D5" target="_self" title="ARYTM_0:def.5">Def5</a></i>;<br/><a NAME="E4:24_1_2"/><i><font color="Green" title="E28">A6</font></i>: 
 ex <font color="Olive" title="b1">x'</font>, <font color="Olive" title="b2">y'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  st <br/>( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y'</font> &amp;  <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Olive" title="b2">y'</font> )
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:24"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="arytm_0.html#D3" target="_self" title="ARYTM_0:def.3">Def3</a>, <a class="ref" href="arytm_2.html#T21" title="ARYTM_2:th.21">ARYTM_2:21</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">x</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> <font color="Maroon" title="c3">x</font></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> <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> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">y</font>
<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> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>
<b>by </b><i><a class="txt" href="arytm_0.html#E1:24"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="arytm_0.html#E1:24_1_2"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="arytm_0.html#D5" target="_self" title="ARYTM_0:def.5">Def5</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="arytm_0.html#E1:24"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="arytm_0.html#E4:24_1_2"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="arytm_2.html#T16" title="ARYTM_2:th.16">ARYTM_2:16</a></i>
;<br/><b>hence </b><a NAME="E6:24_1_2"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E2:24_1_2"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="arytm_0.html#E3:24_1_2"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="arytm_0.html#T20" target="_self" title="ARYTM_0:th.20">Th20</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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