reserve V for non empty set,
  A,B,A9,B9 for Element of V;
reserve f,f9 for Element of Funcs(V);
reserve m,m1,m2,m3,m9 for Element of Maps V;

theorem Th10:
  for f being Function, A,B being set st [[A,B],f] in Maps(V)
  holds (B = {} implies A = {}) & f is Function of A,B
proof
  let f be Function, A,B be set;
  assume [[A,B],f] in Maps(V);
  then consider f9,A9,B9 such that
A1: [[A,B],f] = [[A9,B9],f9] and
A2: ( B9 = {} implies A9 = {})& f9 is Function of A9,B9 by Th4;
  f = f9 & A = A9 by A1,Lm1,XTUPLE_0:1;
  hence thesis by A1,A2,Lm1;
end;
