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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_3.html#K7" target="_self">RAT+</a> , <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a> ;<br/>

<b>set </b><font color="Maroon">nx</font> =  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">x</font>;<br/>
<b>set </b><font color="Maroon">ny</font> =  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font>;<br/>
<b>set </b><font color="Maroon">dx</font> =  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font>;<br/>
<b>set </b><font color="Maroon">dy</font> =  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font>;<br/>
<a NAME="E1:87_1_1"/>
(  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  &amp;  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E2:87_1_1"/><b>then </b><i><font color="Green">E13</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T16" target="_self">Th16</a>, <a class="ref" href="ordinal3.html#T34">ORDINAL3:34</a></i>;<br/>
<a NAME="E3:87_1_1"/><i><font color="Green">E14</font></i>: 
( <span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E4:87_1_1"/>
( <span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span> <a href="ordinal1.html#R1">c=</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font></span>)</span> or <span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font></span>)</span> <a href="ordinal1.html#R1">c=</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span> )
 
;<br/>
<a NAME="E5:87_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="arytm_3.html#K2" target="_self">+^</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" target="_self">numerator</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="ordinal3.html#K5">-^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> or <span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="arytm_3.html#K2" target="_self">+^</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" target="_self">numerator</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="ordinal3.html#K5">-^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="ordinal3.html#D6">ORDINAL3:def 6</a></i>;<br/>
<a NAME="E6:87_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="arytm_3.html#K11" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p4">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="ordinal3.html#K5">-^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p4">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="arytm_3.html#K11" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p4">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="ordinal3.html#K5">-^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p4">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T23" target="_self">Th23</a>, <a class="ref" href="arytm_3.html#T24" target="_self">Th24</a>, </i>;<br/>
<b>hence </b><a NAME="E7:87_1_1"/>
( ( for <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a>   holds not <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Olive">z</font> ) implies  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Olive">z</font> )
 ;<br/>


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