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<b>let </b><font color="Maroon" title="c2">p</font> be   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a>;<br/>

<b>assume </b><a NAME="E1:50"/>
 ex <font color="Olive" title="b1">l</font> being   <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>  st <br/>( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">l</font> &amp;  ex <font color="Olive" title="b2">m</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ex <font color="Olive" title="b3">k</font> being   <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>  st <br/>(  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">l</font> &amp;  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b1">l</font> ) )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">l</font> being    <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> such that </b><br/><a NAME="E3:50"/><i><font color="Green" title="E32">A1</font></i>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">l</font> &amp;  ex <font color="Olive" title="b1">m</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ex <font color="Olive" title="b2">k</font> being   <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>  st <br/>(  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">l</font> &amp;  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">l</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon" title="c4">m</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>, <font color="Maroon" title="c5">k</font> being    <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> such that </b><br/><a NAME="E5:50"/><i><font color="Green" title="E33">A2</font></i>: 
(  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">l</font> &amp;  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">l</font> )
 <b>by </b><i><a class="txt" href="rat_1.html#E3:50"><i><font color="Green" title="E32">A1</font></i></a></i>;<br/>
<a NAME="E6:50"/><i><font color="Green" title="E34">A3</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="rat_1.html#E5:50"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="rat_1.html#D3" target="_self" title="RAT_1:def.3">Def3</a></i>;<br/>
<div><i><font color="Green" title="E35">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>now </b></a><div class="add"><a NAME="E1:50_1"/>
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">l</font>
 <b>by </b><i><a class="txt" href="rat_1.html#E3:50"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="rat_1.html#E6:50"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E2:50_1"/>
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="rat_1.html#E5:50"><i><font color="Green" title="E33">A2</font></i></a></i>;<br/></div><b>end;</b></div>
<font color="Maroon" title="c2">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">l</font></span>)</span>
<b>by </b><i><a class="txt" href="rat_1.html#E5:50"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="rat_1.html#T37" target="_self" title="RAT_1:th.37">Th37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c3">l</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T77" title="XCMPLX_1:th.77">XCMPLX_1:77</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
<b>by </b><i><a class="txt" href="rat_1.html#E3:50"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">k</font>

;<br/>
<b>hence </b><a NAME="E9:50"/>
contradiction
 <b>by </b><i><a class="txt" href="rat_1.html#E6:50"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="rat_1.html#E7:50"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="rat_1.html#D3" target="_self" title="RAT_1:def.3">Def3</a></i>;<br/>


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