reserve G for strict Group,
  a,b,x,y,z for Element of G,
  H,K for strict Subgroup of G,
  p for Element of NAT,
  A for Subset of G;
reserve G for Group;
reserve H, B, A for Subgroup of G,
  D for Subgroup of A;

theorem Th23:
  for G being strict Group st G <> (1).G ex b being Element of G st b <> 1_G
proof
  let G be strict Group such that
A1: G <> (1).G;
  assume
A2: not ex b being Element of G st b <> 1_G;
  for x,y being Element of G holds x = y
  proof
    let x,y be Element of G;
    x = 1_G by A2;
    hence thesis by A2;
  end;
  then G is trivial;
  hence contradiction by A1,GROUP_6:12;
end;
