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<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:24"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</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">x</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </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">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_1_1"/>
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><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">E16</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><div><i><font color="Green">E23</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">E24</font></i>: 
 <a href="arytm_0.html#K4" target="_self">inv</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>thus </b>1 = 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" target="_self">inv</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T6" target="_self">Th6</a>, <a class="txt" href="arytm_0.html#E1:24"><i><font color="Green">E13</font></i></a></i>
<br/>.= 
0
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div><a NAME="E3:24_1_2"/><i><font color="Green">E35</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" target="_self">inv</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="arytm_0.html#T6" target="_self">Th6</a>, <a class="txt" href="arytm_0.html#E1:24"><i><font color="Green">E13</font></i></a></i>;<br/><a NAME="E4:24_1_2"/><i><font color="Green">E36</font></i>: 
 ex <font color="Olive">x'</font>, <font color="Olive">y'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y'</font> &amp;  <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">y</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Olive">y'</font> )
 <b>by </b><i>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a>, <a class="ref" href="arytm_2.html#T21">ARYTM_2:21</a></i>;<br/> <a href="arytm_0.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" target="_self">inv</a> <font color="Maroon">x</font></span>)</span> = 
 <a href="arytm_0.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4" target="_self">inv</a> <font color="Maroon">x</font></span>)</span></span>)</span>,<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T4" target="_self">Th4</a></i>
<br/>.= 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">y</font>,<font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="arytm_0.html#T6" target="_self">Th6</a>, <a class="txt" href="arytm_0.html#E1:24"><i><font color="Green">E13</font></i></a></i>
<br/>.= 
1
<b>by </b><i>, , <a class="ref" href="arytm_2.html#T16">ARYTM_2:16</a></i>
;<br/><b>hence </b><a NAME="E6:24_1_2"/>
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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