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

theorem
  for G being finite Group, H being Subgroup of G holds
  card H <> card G implies ex a being Element of G st not a in H
proof
  let G be finite Group;
  let H be Subgroup of G;
  assume
A1: card H <> card G;
  assume not ex a being Element of G st not a in H;
  then the multMagma of H = the multMagma of G by GROUP_2:62;
  hence contradiction by A1;
end;
