let a, b be real number ; :: thesis: for n being Element of NAT st a >= 0 & a < b & n >= 1 holds
n -Root a < n -Root b
let n be Element of NAT ; :: thesis: ( a >= 0 & a < b & n >= 1 implies n -Root a < n -Root b )
assume that
A1: a >= 0 and
A2: a < b and
A3: n >= 1 and
A4: n -Root a >= n -Root b ; :: thesis: contradiction
(n -Root a) |^ n = a by A1, A3, Th28;
then A5: (n -Root a) |^ n < (n -Root b) |^ n by A1, A2, A3, Lm2;
n -Root b > 0 by A1, A2, A3, Def3;
hence contradiction by A4, A5, Th17; :: thesis: verum