reserve x,y,y1,y2 for set;
reserve G for Group;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;

theorem
  G is commutative Group iff the multF of G is commutative
proof
  thus G is commutative Group implies the multF of G is commutative
  proof
    assume
A1: G is commutative Group;
    let a,b;
    thus (the multF of G).(a,b) = a * b .= b * a by A1,Lm1
      .= (the multF of G).(b,a);
  end;
  assume
A2: the multF of G is commutative;
  G is commutative
  by A2;
  hence thesis;
end;
