let a be real number ; :: thesis: for n being Element of NAT st a >= 0 & n >= 1 holds
n -root a >= 0
let n be Element of NAT ; :: thesis: ( a >= 0 & n >= 1 implies n -root a >= 0 )
assume A1: ( a >= 0 & n >= 1 ) ; :: thesis: n -root a >= 0

now

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

suppose A2: a > 0 ; :: thesis: n -root a >= 0

n -root a = n -Root a by A1, Def1;
hence n -root a >= 0 by A1, A2, PREPOWER:def 3; :: thesis: verum

end;

suppose A3: a = 0 ; :: thesis: n -root a >= 0

n -root a = n -Root a by A1, Def1
.= 0 by A1, A3, PREPOWER:def 3 ;
hence n -root a >= 0 ; :: thesis: verum

end;

end;

end;

hence n -root a >= 0 ; :: thesis: verum