theorem Th25:
  G is commutative Group implies A * B = B * A
proof
  assume
A1: G is commutative Group;
  thus A * B c= B * A
  proof
    let x be object;
    assume x in A * B;
    then consider g,h such that
A2: x = g * h and
A3: g in A & h in B;
    x = h * g by A1,A2,Lm2;
    hence thesis by A3;
  end;
  let x be object;
  assume x in B * A;
  then consider g,h such that
A4: x = g * h and
A5: g in B & h in A;
  x = h * g by A1,A4,Lm2;
  hence thesis by A5;
end;
