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<span class="kw">let </span><font color="Maroon" title="c3">x</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> ; <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">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  implies  ex <font color="Olive" title="b1">y</font> being    <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>  st <font color="Maroon" title="c3">x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">nx</font> =  <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c3">x</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">dx</font> =  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c3">x</font>;<br/>
<a NAME="E1:69"/><span class="lab"><font color="Green" title="E62">A1</font></span>: 
 <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T35" target="_self" title="ARYTM_3:th.35">Th41</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:69"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <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>  st <font color="Maroon" title="c3">x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<a NAME="E3:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A2</font></span>: 
 <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T37" target="_self" title="ARYTM_3:th.37">Th43</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c6">y</font> = <span class="p1">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c3">x</font></span>)</span>; <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">x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<a NAME="E4:69"/>
 <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c3">x</font>, <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c3">x</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T34" target="_self" title="ARYTM_3:th.34">Th40</a></span>;<br/>
<a NAME="E5:69"/><span class="kw">then </span>
(  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c3">x</font> &amp;  <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c3">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="ref" href="arytm_3.html#T43" target="_self" title="ARYTM_3:th.43">Th49</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:69"/>
<font color="Maroon" title="c3">x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="ref" href="arytm_3.html#T41" target="_self" title="ARYTM_3:th.41">Th47</a>, <a class="ref" href="ordinal3.html#T31" title="ORDINAL3:th.31">ORDINAL3:31</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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