let a be real number ; :: thesis: for n being Element of NAT st a >= 0 & n >= 1 holds
a to_power (1 / n) = n -root a
let n be Element of NAT ; :: thesis: ( a >= 0 & n >= 1 implies a to_power (1 / n) = n -root a )
assume that
A1: a >= 0 and
A2: n >= 1 ; :: thesis: a to_power (1 / n) = n -root a
reconsider p = n " as Rational ;

now

per cases ( a > 0 or a = 0 ) by A1;

suppose a > 0 ; :: thesis: a to_power (1 / n) = n -Root a

hence a to_power (1 / n) = a #Q p by Th51
.= n -Root a by A2, PREPOWER:50 ;
:: thesis: verum

end;

suppose A3: a = 0 ; :: thesis: a to_power (1 / n) = n -Root a

hence a to_power (1 / n) = 0 by A2, Def2
.= n -Root a by A2, A3, PREPOWER:def 2 ;
:: thesis: verum

end;

end;

end;

hence a to_power (1 / n) = n -root a by A1, A2, Def1; :: thesis: verum