
theorem Th14:
  for S being up-complete lower-bounded LATTICE,
  T being continuous lower-bounded LATTICE, f being Function of S, T holds
  ( ( for x being Element of S holds
  f.x = "\/"({ f.w where w is Element of S : w << x },T) ) implies
  for x being Element of S, y being Element of T
  holds y << f.x iff ex w being Element of S st w << x & y << f.w )
proof
  let S be up-complete lower-bounded LATTICE;
  let T be continuous lower-bounded LATTICE;
  let f be Function of S, T;
  assume
A1: for x being Element of S holds
  f.x = "\/"({ f.w where w is Element of S : w << x },T);
  then
A2: f is monotone by Th13;
  let x be Element of S, y be Element of T;
  hereby
    assume
A3: y << f.x;
    reconsider D = f.:(waybelow x) as non empty directed Subset of T
    by A1,Th13,YELLOW_2:15;
A4: f.x = "\/"({ f.w where w is Element of S : w << x },T) by A1;
A5: the carrier of S c= dom f by FUNCT_2:def 1;
    defpred P[Element of S] means $1 << x;
    deffunc A(Element of S) = $1;
    f.:{ A(w) where w is Element of S : P[w] } =
    { f.A(w) where w is Element of S : P[w] } from FuncFraenkelSL(A5);
    then consider w being Element of T such that
A6: w in D and
A7: y << w by A3,A4,WAYBEL_4:53;
    consider v be object such that
A8: v in the carrier of S and
A9: v in waybelow x and
A10: w = f.v by A6,FUNCT_2:64;
    reconsider v as Element of S by A8;
    take v;
    thus v << x & y << f.v by A7,A9,A10,WAYBEL_3:7;
  end;
  given w being Element of S such that
A11: w << x and
A12: y << f.w;
  w <= x by A11,WAYBEL_3:1;
  then f.w <= f.x by A2;
  hence thesis by A12,WAYBEL_3:2;
end;
