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<b>let </b><font color="Maroon" title="c1">q</font> be   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a>;<br/>

<b>set </b><font color="Maroon" title="c2">k</font> =  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c1">q</font>;<br/>
<b>set </b><font color="Maroon" title="c3">h</font> =  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c1">q</font>;<br/>
<a NAME="E1:55"/><i><font color="Green" title="E40">A1</font></i>: 
 <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c1">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="rat_1.html#D3" title="RAT_1:def.3">RAT_1:def 3</a></i>;<br/>
<b>assume </b><a NAME="E2:55"/><i><font color="Green" title="E41">A2</font></i>: 
not  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c1">q</font>, <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c1">q</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">a</font> = <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c1">q</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c1">q</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_2.html#T29" title="INT_2:th.29">INT_2:29</a></i>;<br/>
<a NAME="E4:55"/><i><font color="Green" title="E42">A3</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E2:55"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></i>;<br/>
<a NAME="E5:55"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E1:55"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="int_2.html#T35" title="INT_2:th.35">INT_2:35</a></i>;<br/>
<a NAME="E6:55"/><b>then </b>
<font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E7:55"/><b>then </b>
<font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E8:55"/><b>then </b><i><font color="Green" title="E43">A4</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E4:55"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E9:55"/><i><font color="Green" title="E44">A5</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c1">q</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T31" title="INT_2:th.31">INT_2:31</a></i>;<br/>
<a NAME="E10:55"/><i><font color="Green" title="E45">A6</font></i>: 
<span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c1">q</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c1">q</font></span>)</span> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c1">q</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T32" title="INT_2:th.32">INT_2:32</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">l</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E12:55"/><i><font color="Green" title="E46">A7</font></i>: 
 <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">l</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E9:55"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">b</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E14:55"/><i><font color="Green" title="E47">A8</font></i>: 
 <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E10:55"><i><font color="Green" title="E45">A6</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c6">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E16:55"/>
 <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">b</font>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E14:55"><i><font color="Green" title="E47">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:55"/>
contradiction
 <b>by </b><i><a class="txt" href="wsierp_1.html#E8:55"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="wsierp_1.html#E12:55"><i><font color="Green" title="E46">A7</font></i></a>, <a class="ref" href="rat_1.html#T62" title="RAT_1:th.62">RAT_1:62</a></i>;<br/>


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