theorem
  G is_distributive_wrt F & u = G[;](d,id D) implies u is_distributive_wrt F
proof
  assume that
A1: G is_distributive_wrt F and
A2: u = G[;](d,id D);
  let d1,d2;
  thus u.(F.(d1,d2)) = G.(d,(id D).(F.(d1,d2))) by A2,FUNCOP_1:53
    .= G.(d,F.(d1,d2))
    .= F.(G.(d,d1),G.(d,d2)) by A1,BINOP_1:11
    .= F.(G.(d,(id D).d1),G.(d,d2))
    .= F.(G.(d,(id D).d1),G.(d,(id D).d2))
    .= F.(u.d1,G.(d,(id D).d2)) by A2,FUNCOP_1:53
    .= F.(u.d1,u.d2) by A2,FUNCOP_1:53;
end;
