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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</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>assume </b><b>that </b><br/><a NAME="E1:76"/><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>and </b><br/><a NAME="E2:76"/><i><font color="Green" title="E46">A2</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 ;<br/>

<a NAME="E3:76"/><i><font color="Green" title="E47">A3</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></i>;<br/>
<a NAME="E4:76"/>
( <font color="Maroon" title="c1">n</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c1">n</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 )
 
<b>by </b><i><a class="txt" href="newton.html#E2:76"><i><font color="Green" title="E46">A2</font></i></a></i>;<br/>
<a NAME="E5:76"/><b>then </b>
 ex <font color="Olive" title="b1">r</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">r</font>
 
<b>by </b><i><a class="txt" href="newton.html#E3:76"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:76"/>
contradiction
 <b>by </b><i><a class="txt" href="newton.html#E1:76"><i><font color="Green" title="E45">A1</font></i></a></i>;<br/>


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