
theorem Th38:
  for X being non empty set, S,T being non empty Poset for f being
  directed-sups-preserving Function of S, T|^X for i being Element of X holds (
  commute f).i is directed-sups-preserving Function of S, T
proof
  let M be non empty set, X,Y be non empty Poset;
  let f be directed-sups-preserving Function of X, Y|^M;
  let i be Element of M;
A1: (M --> Y).i = Y;
  dom commute f = M by Lm1;
  then
A2: (commute f).i in rng commute f by FUNCT_1:def 3;
A3: f in Funcs(the carrier of X, Funcs(M, the carrier of Y)) by Lm1;
  then f is Function of the carrier of X, Funcs(M, the carrier of Y) by
FUNCT_2:66;
  then
A4: rng f c= Funcs(M, the carrier of Y) by RELAT_1:def 19;
  rng commute f c= Funcs(the carrier of X, the carrier of Y) by Lm1;
  then consider g being Function such that
A5: (commute f).i = g and
A6: dom g = the carrier of X and
A7: rng g c= the carrier of Y by A2,FUNCT_2:def 2;
  reconsider g as Function of X,Y by A6,A7,FUNCT_2:2;
A8: Y|^M = product (M --> Y) by YELLOW_1:def 5;
  g is directed-sups-preserving
  proof
    let A be Subset of X;
    assume A is non empty directed;
    then reconsider B = A as non empty directed Subset of X;
    assume
A9: ex_sup_of A, X;
A10: f preserves_sup_of B by WAYBEL_0:def 37;
    then
A11: ex_sup_of f.:B, Y|^M by A9;
    then ex_sup_of pi(f.:A, i), Y by A8,A1,YELLOW16:31;
    hence ex_sup_of g.:A, Y by A4,A5,Th8;
A12: pi(f.:A, i) = g.:A by A4,A5,Th8;
    sup (f.:B) = f.sup B by A9,A10;
    then sup pi(f.:A, i) = (f.sup A).i by A11,A8,A1,YELLOW16:33;
    hence thesis by A3,A5,A12,FUNCT_6:56;
  end;
  hence thesis by A5;
end;
