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<b>let </b><font color="Maroon" title="c2">k</font> be    <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> ;<br/><b>let </b><font color="Maroon" title="c3">m</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/><b>let </b><font color="Maroon" title="c4">p</font> be   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a>;<br/>





<b>assume </b><a NAME="E1:51"/><i><font color="Green" title="E33">A1</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">k</font> &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; ( for <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>   holds <br/>(  not 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">l</font> or for <font color="Olive" title="b2">m1</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/> for <font color="Olive" title="b3">k1</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>   holds <br/>(  not <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">l</font> or  not <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">k1</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="c5">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="E3:51"/><i><font color="Green" title="E34">A2</font></i>: 
( <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">l</font> &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c5">l</font> )
 <b>by </b><i><a class="ref" href="rat_1.html#T60" target="_self" title="RAT_1:th.60">Th60</a></i>;<br/>
<a NAME="E4:51"/><i><font color="Green" title="E35">A3</font></i>: 
<font color="Maroon" title="c5">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="rat_1.html#E1:51"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="rat_1.html#E3:51"><i><font color="Green" title="E34">A2</font></i></a></i>;<br/>
<a NAME="E5:51"/>
<font color="Maroon" title="c5">l</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#E1:51"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="rat_1.html#E3:51"><i><font color="Green" title="E34">A2</font></i></a></i>;<br/>
<a NAME="E6:51"/><b>then </b>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">l</font>
 
;<br/>
<a NAME="E7:51"/><b>then </b>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">l</font>
 
<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="E8:51"/><b>then </b><i><font color="Green" title="E36">A4</font></i>: 
<font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="rat_1.html#E4:51"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E9:51"/>
<font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="rat_1.html#E3:51"><i><font color="Green" title="E34">A2</font></i></a></i>;<br/>

<b>thus </b><a NAME="E10:51"/>
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="rat_1.html#E3:51"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="rat_1.html#E8:51"><i><font color="Green" title="E36">A4</font></i></a></i>;<br/>


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