theorem
  F is associative & F is having_a_unity & F is commutative &
  F is having_an_inverseOp & u = the_inverseOp_wrt F implies
  u.(F*(id D,u).(d1,d2)) =
    F*(u,id D).(d1,d2) & F*(id D,u).(d1,d2) = u.(F*(u,id D).(d1,d2))
proof
  assume that
A1: F is associative & F is having_a_unity and
A2: F is commutative and
A3: F is having_an_inverseOp & u = the_inverseOp_wrt F;
A4: u is_distributive_wrt F by A1,A2,A3,Th63;
  thus u.(F*(id D,u).(d1,d2)) = u.(F.(d1,u.d2)) by Th81
    .= F.(u.d1,u.(u.d2)) by A4
    .= F.(u.d1,d2) by A1,A3,Th62
    .= F*(u,id D).(d1,d2) by Th81;
  hence thesis by A1,A3,Th62;
end;
