theorem
  G is_distributive_wrt F & u = G[:](id D,d) implies u is_distributive_wrt F
proof
  assume that
A1: G is_distributive_wrt F and
A2: u = G[:](id D,d);
  let d1,d2;
  thus u.(F.(d1,d2)) = G.((id D).(F.(d1,d2)),d) by A2,FUNCOP_1:48
    .= G.(F.(d1,d2),d)
    .= F.(G.(d1,d),G.(d2,d)) by A1,BINOP_1:11
    .= F.(G.((id D).d1,d),G.(d2,d))
    .= F.(G.((id D).d1,d),G.((id D).d2,d))
    .= F.(u.d1,G.((id D).d2,d)) by A2,FUNCOP_1:48
    .= F.(u.d1,u.d2) by A2,FUNCOP_1:48;
end;
