reserve a, I for set,
  S for non empty non void ManySortedSign;

theorem
  for X, Y being set, f being Function st a in dom f & f.a in [:X,Y:]
  holds f.a = [(pr1 f).a, (pr2 f).a]
proof
  let X, Y be set, f be Function such that
A1: a in dom f and
A2: f.a in [:X,Y:];
  (pr1 f).a = (f.a)`1 & (pr2 f).a = (f.a)`2 by A1,MCART_1:def 12,def 13;
  hence thesis by A2,MCART_1:21;
end;
