<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="rat_1.html#NM1">Rational</a>;<br/>





<b>assume </b><a NAME="E1:61"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#R1">are_relative_prime</a>  )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:61"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="rat_1.html#T60">RAT_1:60</a></i>;<br/>
<a NAME="E4:61"/>
 <a href="rat_1.html#K2">numerator</a> <font color="Maroon">c<sub>3</sub></font>, <a href="rat_1.html#K1">denominator</a> <font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#R1">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="wsierp_1.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E5:61"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="prepower.html#K1">abs</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E3:61"><i><font color="Green">E44</font></i></a>, <a class="ref" href="int_2.html#T39">INT_2:39</a></i>;<br/>
<b>then </b>1 = 
 <a href="prepower.html#K1">abs</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="wsierp_1.html#E1:61"><i><font color="Green">E43</font></i></a>, <a class="ref" href="int_2.html#D4">INT_2:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma8</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E7:61"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rat_1.html#K2">numerator</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rat_1.html#K1">denominator</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E3:61"><i><font color="Green">E44</font></i></a></i>;<br/>


</div>
