reserve G for Group,
  a,b for Element of G,
  m, n for Nat,
  p for Prime;

theorem Th15:
  for G being finite Group,H being Subgroup of G
  for a being Element of G st H is p-group & a in H
  holds a is p-power
proof
  let G be finite Group;
  let H be Subgroup of G;
  let a be Element of G;
  assume that
A1: H is p-group and
A2: a in H;
 a is Element of H by A2;
  then consider b be Element of H such that
A3: b= a;
  consider r be Nat such that
A4: ord b = p |^ r by A1,Def1;
  ord a = p |^ r by A3,A4,GROUP_8:5;
  hence thesis;
end;
