let a, b be Real; :: thesis: for n being Nat st 0 <= a & a <= b & n >= 1 holds

n -Root a <= n -Root b

let n be Nat; :: thesis: ( 0 <= a & 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, Th19;

then (n -Root a) |^ n <= (n -Root b) |^ n by A1, A2, A3, Th19;

hence contradiction by A3, A4, A5, Th10; :: thesis: verum

n -Root a <= n -Root b

let n be Nat; :: thesis: ( 0 <= a & 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, Th19;

then (n -Root a) |^ n <= (n -Root b) |^ n by A1, A2, A3, Th19;

hence contradiction by A3, A4, A5, Th10; :: thesis: verum