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<b>let </b><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">n</font>, <font color="Maroon" title="c3">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:43"/><i><font color="Green" title="E30">A1</font></i>: 
( <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c1">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">m</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1_1"/>
( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> or <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1"><b>suppose </b></a><a NAME="E1:43_1_1_1"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:43_1_1_1"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_2"><b>suppose </b></a><a NAME="E1:43_1_1_2"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">m</font>
 ;<br/><div class="add"><a NAME="E2:43_1_1_2"/><b>then </b><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="pepin.html#E1:43"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="power.html#T57" title="POWER:th.57">POWER:57</a></i>;<br/><a NAME="E3:43_1_1_2"/>
<font color="Maroon" title="c1">k</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="pepin.html#E1:43"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>;<br/><b>hence </b><a NAME="E4:43_1_1_2"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="txt" href="pepin.html#E1:43"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="pepin.html#E2:43_1_1_2"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:43"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 ;<br/>


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