reserve n for Element of NAT;
reserve i for Integer;
reserve G,H,I for Group;
reserve A,B for Subgroup of G;
reserve N for normal Subgroup of G;
reserve a,a1,a2,a3,b,b1 for Element of G;
reserve c,d for Element of H;
reserve f for Function of the carrier of G, the carrier of H;
reserve x,y,y1,y2,z for set;
reserve A1,A2 for Subset of G;

theorem Th6:
  for G being Group, A1 being Subset of G holds
  A1 = the set of all [.a,b.] where
    a is Element of G, b is Element of G implies G` = gr A1
proof
  let G be Group, A1 be Subset of G;
  assume
A1: A1 = the set of all [.a,b.] where a is Element of G, b is Element of G;
  A1 = commutators G
  proof
    thus A1 c= commutators G
    proof
      let x be object;
      assume x in A1;
      then ex a,b being Element of G st x = [.a,b.] by A1;
      hence thesis by GROUP_5:58;
    end;
    let x be object;
    assume x in commutators G;
    then ex a,b being Element of G st x = [.a,b.] by GROUP_5:58;
    hence thesis by A1;
  end;
  hence thesis by GROUP_5:72;
end;
