
theorem Th24: ::  see YELLOW_2:17, for directed
  for S,T being non empty RelStr, f being Function of S,T st f is
monotone for X being Subset of S holds (X is filtered implies f.:X is filtered)
proof
  let S,T be non empty RelStr, f be Function of S,T;
  assume
A1: f is monotone;
  let X be Subset of S such that
A2: X is filtered;
  let x,y be Element of T;
  assume x in f.:X;
  then consider a being object such that
A3: a in the carrier of S and
A4: a in X and
A5: x = f.a by FUNCT_2:64;
  assume y in f.:X;
  then consider b being object such that
A6: b in the carrier of S and
A7: b in X and
A8: y = f.b by FUNCT_2:64;
  reconsider a,b as Element of S by A3,A6;
  consider c being Element of S such that
A9: c in X and
A10: c <= a & c <= b by A2,A4,A7;
  take z = f.c;
  thus z in f.:X by A9,FUNCT_2:35;
  thus thesis by A1,A5,A8,A10;
end;
