reserve X for set;
reserve UN for Universe;

theorem Th55:
  for UN being non trivial Universe holds COMPLEX in UN
  proof
    let UN be non trivial Universe;
    set X = Funcs({0,1},REAL),
        Y = {x where x is Element of Funcs({0,1},REAL): x.1 = 0};
A1: (X \ Y) \/ REAL c= X \/ REAL by XBOOLE_1:13;
A2: 0 in UN & 1 in UN by Th16;
    UN is axiom_GU2;
    then
A3: {0,1} in UN by A2;
A4: REAL is Element of UN by Th53;
    then Funcs({0,1},REAL) in UN by A3,CLASSES2:61;
    then X \/ REAL in UN by A4,CLASSES2:60;
    hence thesis by Th13,A1;
  end;
