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;

theorem Th7:
  for G being Group for a being Element of G st a <> 1_G holds gr {a} <> (1).G
proof
  let G be Group;
  let a be Element of G such that
A1: a <> 1_G;
  assume
A2: gr {a} = (1).G;
A3: the carrier of (1).G = {1_G} by GROUP_2:def 7;
A4: {a} c= the carrier of gr {a} by GROUP_4:def 4;
  for xx being set holds xx = a implies xx = 1_G
  proof
    let xx be set;
    assume xx = a;
    then xx in {a} by TARSKI:def 1;
    hence thesis by A2,A3,A4,TARSKI:def 1;
  end;
  hence thesis by A1;
end;
