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<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> ;<br/>



<b>assume </b><a NAME="E1:67"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  )
 ;<br/>

<b>then reconsider </b><font color="Maroon">x0</font> = <font color="Maroon">x</font>, <font color="Maroon">y0</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a>  ;<br/>
<b>consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Maroon">y'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E4:67"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 <b>and </b><br/><a NAME="E5:67"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 <b>and </b><br/><a NAME="E6:67"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <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_2.html#D5" target="_self">Def5</a>, , <a class="ref" href="arytm_3.html#T34">ARYTM_3:34</a></i>;<br/>
<a NAME="E7:67"/>
<font color="Maroon">x'</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> <a href="arytm_3.html#K2">+^</a> <font color="Maroon">y0</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E24"><i><font color="Green">Lemma62</font></i></a>, <a class="txt" href="arytm_2.html#E25"><i><font color="Green">Lemma63</font></i></a>, <a class="txt" href="arytm_2.html#E22"><i><font color="Green">Lemma58</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:67"/>
<font color="Maroon">y</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i>, <a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>


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