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<span class="kw">let </span><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a>  &amp; <font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies (  <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> &amp;  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:55"/>
<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or (  <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> &amp;  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> ) )</span><br/>

<a NAME="E2:55"/><span class="kw">then </span>
(  <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> &amp;  <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> (<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T23" target="_self" title="ARYTM_3:th.23">Th28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:55"/>
(  not <font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or (  <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> &amp;  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T42" target="_self" title="ARYTM_3:th.42">Th48</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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