reserve x,y for set,
  k,n for Nat,
  i for Integer,
  G for Group,
  a,b,c ,d,e for Element of G,
  A,B,C,D for Subset of G,
  H,H1,H2,H3,H4 for Subgroup of G ,
  N1,N2 for normal Subgroup of G,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

theorem
  for G being strict Group holds G is commutative Group iff center G = G
proof
  let G be strict Group;
  thus G is commutative Group implies center G = G
  proof
    assume
A1: G is commutative Group;
    now
      let a be Element of G;
      for b being Element of G holds a * b = b * a by A1,Lm1;
      hence a in center G by Th77;
    end;
    hence thesis by GROUP_2:62;
  end;
  assume
A2: center G = G;
  G is commutative
  by STRUCT_0:def 5,A2,Th77;
  hence thesis;
end;
