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<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a> , <font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a> ;<br/><b>let </b><font color="Maroon">x'</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> ;<br/><b>let </b><font color="Maroon">y'</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> ;<br/>





<b>assume </b><a NAME="E1:64"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 ;<br/>

<a NAME="E2:64"/><b>then </b><i><font color="Green">E20</font></i>: 
 <a href="arytm_3.html#K9">denominator</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K14">one</a> 
 
<b>by </b><i><a class="ref" href="arytm_3.html#D9">ARYTM_3:def 9</a></i>;<br/>
<b>assume </b><a NAME="E3:64"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 ;<br/>

<a NAME="E4:64"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="arytm_3.html#K9">denominator</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K14">one</a> 
 
<b>by </b><i><a class="ref" href="arytm_3.html#D9">ARYTM_3:def 9</a></i>;<br/>
<b>set </b><font color="Maroon">a</font> = <span class="p1">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">x'</font></span>)</span> <a href="arytm_3.html#K3">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">y'</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">b</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K8">numerator</a> <font color="Maroon">x'</font></span>)</span> <a href="arytm_3.html#K3">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">y'</font></span>)</span></span>)</span> <a href="arytm_3.html#K2">+^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K8">numerator</a> <font color="Maroon">y'</font></span>)</span> <a href="arytm_3.html#K3">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">x'</font></span>)</span></span>)</span>;<br/>
<a NAME="E5:64"/><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">x'</font></span>)</span> <a href="arytm_3.html#K3">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">y'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K14">one</a> 
 
<b>by </b><i>, <a class="txt" href="arytm_2.html#E24"><i><font color="Green">Lemma62</font></i></a>, <a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>;<br/>
<i><font color="Green">E33</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K8">numerator</a> <font color="Maroon">x'</font></span>)</span> <a href="arytm_3.html#K3">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">y'</font></span>)</span></span>)</span> <a href="arytm_3.html#K2">+^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K8">numerator</a> <font color="Maroon">y'</font></span>)</span> <a href="arytm_3.html#K3">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">x'</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K8">numerator</a> <font color="Maroon">x'</font></span>)</span> <a href="arytm_3.html#K3">*^</a> <a href="arytm_3.html#K14">one</a> </span>)</span> <a href="arytm_3.html#K2">+^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K8">numerator</a> <font color="Maroon">y'</font></span>)</span>
<b>by </b><i>, <a class="txt" href="arytm_2.html#E24"><i><font color="Green">Lemma62</font></i></a>, <a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="arytm_3.html#K8">numerator</a> <font color="Maroon">x'</font></span>)</span> <a href="arytm_3.html#K2">+^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K8">numerator</a> <font color="Maroon">y'</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>
<br/>.= 
<font color="Maroon">i</font> <a href="arytm_3.html#K2">+^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K8">numerator</a> <font color="Maroon">y'</font></span>)</span>
<b>by </b><i>, <a class="ref" href="arytm_3.html#D8">ARYTM_3:def 8</a></i>
<br/>.= 
<font color="Maroon">i</font> <a href="arytm_3.html#K2">+^</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="arytm_2.html#D5" target="_self">Def5</a>, <a class="ref" href="arytm_3.html#D8">ARYTM_3:def 8</a></i>
;<br/>
<a NAME="E7:64"/><i><font color="Green">E37</font></i>: 
 <a href="arytm_3.html#K6">RED</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">x'</font></span>)</span> <a href="arytm_3.html#K3">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">y'</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K8">numerator</a> <font color="Maroon">x'</font></span>)</span> <a href="arytm_3.html#K3">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">y'</font></span>)</span></span>)</span> <a href="arytm_3.html#K2">+^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K8">numerator</a> <font color="Maroon">y'</font></span>)</span> <a href="arytm_3.html#K3">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">x'</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K14">one</a> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E25"><i><font color="Green">Lemma63</font></i></a>, <a class="ref" href="arytm_3.html#T29">ARYTM_3:29</a></i>;<br/>
<b>thus </b><font color="Maroon">i</font> <a href="arytm_3.html#K2">+^</a> <font color="Maroon">j</font> = 
 <a href="arytm_3.html#K6">RED</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K8">numerator</a> <font color="Maroon">x'</font></span>)</span> <a href="arytm_3.html#K3">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">y'</font></span>)</span></span>)</span> <a href="arytm_3.html#K2">+^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K8">numerator</a> <font color="Maroon">y'</font></span>)</span> <a href="arytm_3.html#K3">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">x'</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">x'</font></span>)</span> <a href="arytm_3.html#K3">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">y'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="arytm_2.html#E25"><i><font color="Green">Lemma63</font></i></a>, , <a class="ref" href="arytm_3.html#T29">ARYTM_3:29</a></i>
<br/>.= 
<font color="Maroon">x'</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">y'</font>
<b>by </b><i>, <a class="ref" href="arytm_3.html#D10">ARYTM_3:def 10</a></i>
;<br/>


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