<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ;<br/><b>let </b><font color="Maroon" title="c6">x'</font>, <font color="Maroon" title="c7">y'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> ;<br/>





<b>assume </b><a NAME="E1:64"/><i><font color="Green" title="E59">A1</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font>
 ;<br/>

<a NAME="E2:64"/><b>then </b><i><font color="Green" title="E60">A2</font></i>: 
 <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c6">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 
<b>by </b><i><a class="ref" href="arytm_3.html#D9" title="ARYTM_3:def.9">ARYTM_3:def 9</a></i>;<br/>
<b>assume </b><a NAME="E3:64"/><i><font color="Green" title="E61">A3</font></i>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font>
 ;<br/>

<a NAME="E4:64"/><b>then </b><i><font color="Green" title="E62">A4</font></i>: 
 <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c7">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 
<b>by </b><i><a class="ref" href="arytm_3.html#D9" title="ARYTM_3:def.9">ARYTM_3:def 9</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">a</font> = <span class="p1">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c6">x'</font></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c7">y'</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">b</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c6">x'</font></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c7">y'</font></span>)</span></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c7">y'</font></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c6">x'</font></span>)</span></span>)</span>;<br/>
<a NAME="E5:64"/><i><font color="Green" title="E63">A5</font></i>: 
<span class="p1">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c6">x'</font></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c7">y'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E2:64"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="arytm_2.html#E4:64"><i><font color="Green" title="E62">A4</font></i></a>, <a class="ref" href="ordinal2.html#T56" title="ORDINAL2:th.56">ORDINAL2:56</a></i>;<br/>
<i><font color="Green" title="E64">A6</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c6">x'</font></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c7">y'</font></span>)</span></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c7">y'</font></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c6">x'</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c6">x'</font></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> </span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c7">y'</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_2.html#E2:64"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="arytm_2.html#E4:64"><i><font color="Green" title="E62">A4</font></i></a>, <a class="ref" href="ordinal2.html#T56" title="ORDINAL2:th.56">ORDINAL2:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c6">x'</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c7">y'</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T56" title="ORDINAL2:th.56">ORDINAL2:56</a></i>
<br/>.= 
<font color="Maroon" title="c4">i</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c7">y'</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_2.html#E1:64"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="arytm_3.html#D8" title="ARYTM_3:def.8">ARYTM_3:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">i</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c5">j</font>
<b>by </b><i><a class="txt" href="arytm_2.html#E3:64"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="arytm_3.html#D8" title="ARYTM_3:def.8">ARYTM_3:def 8</a></i>
;<br/>
<a NAME="E7:64"/><i><font color="Green" title="E65">A7</font></i>: 
 <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c6">x'</font></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c7">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#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c6">x'</font></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c7">y'</font></span>)</span></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c7">y'</font></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c6">x'</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E5:64"><i><font color="Green" title="E63">A5</font></i></a>, <a class="ref" href="arytm_3.html#T29" title="ARYTM_3:th.29">ARYTM_3:29</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">i</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c5">j</font> = 
 <a href="arytm_3.html#K4" title="ARYTM_3:func.4">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#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c6">x'</font></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c7">y'</font></span>)</span></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c7">y'</font></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c6">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#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c6">x'</font></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c7">y'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="arytm_2.html#E5:64"><i><font color="Green" title="E63">A5</font></i></a>, <a class="txt" href="arytm_2.html#E6:64"><i><font color="Green" title="E64">A6</font></i></a>, <a class="ref" href="arytm_3.html#T29" title="ARYTM_3:th.29">ARYTM_3:29</a></i>
<br/>.= 
<font color="Maroon" title="c6">x'</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c7">y'</font>
<b>by </b><i><a class="txt" href="arytm_2.html#E7:64"><i><font color="Green" title="E65">A7</font></i></a>, <a class="ref" href="arytm_3.html#D10" title="ARYTM_3:def.10">ARYTM_3:def 10</a></i>
;<br/>


</div>
