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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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"><span class="kw">suppose </span></a><a NAME="E1:27_1_1"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
( <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A2</font></span>: 
(  <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>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#D4" target="_self" title="ARYTM_0:def.4">Def5</a></span>;<br/><a NAME="E3:27_1_1"/>
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_1"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="arytm_0.html#T12" target="_self" title="ARYTM_0:th.12">Th14</a></span>;<br/><span class="kw">hence </span> <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> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#D4" target="_self" title="ARYTM_0:def.4">Def5</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#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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1_1"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="arytm_0.html#T12" target="_self" title="ARYTM_0:th.12">Th14</a></span>
;<br/> <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:27_1_2"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E2:27_1_2"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_2"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E2:27_1_2"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="arytm_0.html#T21" target="_self" title="ARYTM_0:th.21">Th23</a></span>;<br/><a NAME="E4:27_1_2"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
 <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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_2"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="arytm_0.html#D4" target="_self" title="ARYTM_0:def.4">Def5</a></span>;<br/><a NAME="E5:27_1_2"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
 <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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1_2"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="arytm_0.html#D4" target="_self" title="ARYTM_0:def.4">Def5</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">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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T13" target="_self" title="ARYTM_0:th.13">Th15</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="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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27_1_2"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="arytm_0.html#T13" target="_self" title="ARYTM_0:th.13">Th15</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27_1_2"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="arytm_0.html#T19" target="_self" title="ARYTM_0:th.19">Th21</a></span>
;<br/><span class="kw">hence </span><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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_1_2"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="ref" href="arytm_0.html#D4" target="_self" title="ARYTM_0:def.4">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></div><span class="kw">end;</span></div>

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