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<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<b>reconsider </b><font color="Maroon" title="c4">x1</font> = <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c5">y1</font> = <font color="Maroon" title="c3">y</font> as    <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>  <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E2:7"/><i><font color="Green" title="E5">A1</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="txt" href="axioms.html#E23"><i><font color="Green" title="E3">Lm3</font></i></a></i>;<br/>
<b>assume </b><a NAME="E3:7"/><i><font color="Green" title="E6">A2</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 ;<br/>

<a NAME="E4:7"/><b>then </b>
 ex <font color="Olive" title="b1">x'</font>, <font color="Olive" title="b2">y'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  st <br/>( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Maroon" title="c5">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y'</font> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Olive" title="b2">y'</font> )
 
<b>by </b><i><a class="ref" href="arytm_0.html#D2" title="ARYTM_0:def.2">ARYTM_0:def 2</a>, <a class="ref" href="arytm_2.html#T2" title="ARYTM_2:th.2">ARYTM_2:2</a></i>;<br/>
<b>hence </b><a NAME="E5:7"/>
<font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="axioms.html#E2:7"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="axioms.html#E3:7"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="arytm_2.html#T17" title="ARYTM_2:th.17">ARYTM_2:17</a></i>;<br/>


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