let a, b be real number ; :: thesis: for n being natural number st 0 < a & a <= b holds
a |^ n <= b |^ n
let n be natural number ; :: thesis: ( 0 < a & a <= b implies a |^ n <= b |^ n )
assume that
A1: 0 < a and
A2: a <= b ; :: thesis: a |^ n <= b |^ n
defpred S₁[ natural number ] means a |^ $1 <= b |^ $1;
A3: for m1 being natural number st S₁[m1] holds
S₁[m1 + 1]

proof

let m1 be natural number ; :: thesis: ( S₁[m1] implies S₁[m1 + 1] )
assume A4: a |^ m1 <= b |^ m1 ; :: thesis: S₁[m1 + 1]
a |^ m1 > 0 by A1, Th13;
then (a |^ m1) * a <= (b |^ m1) * b by A1, A2, A4, XREAL_1:66;
then a |^ (m1 + 1) <= (b |^ m1) * b by NEWTON:6;
hence S₁[m1 + 1] by NEWTON:6; :: thesis: verum

end;

A5: b |^ 0 = (b GeoSeq) . 0 by Def1
.= 1 by Th4 ;
a |^ 0 = (a GeoSeq) . 0 by Def1
.= 1 by Th4 ;
then A6: S₁[ 0 ] by A5;
for m1 being natural number holds S₁[m1] from NAT_1:sch 2(A6, A3);
hence a |^ n <= b |^ n ; :: thesis: verum