
theorem Th69:
  for S,T being non empty Poset, f being Function of S,T st
  for X being Filter of S holds f preserves_inf_of X holds f is monotone
proof
  let S,T be non empty Poset, f be Function of S,T;
  assume
A1: for X being Filter of S holds f preserves_inf_of X;
  let x,y be Element of S;
A2: ex_inf_of {x}, S by YELLOW_0:38;
A3: ex_inf_of {y}, S by YELLOW_0:38;
A4: f preserves_inf_of uparrow x by A1;
A5: f preserves_inf_of uparrow y by A1;
A6: ex_inf_of uparrow x, S by A2,Th37;
A7: ex_inf_of uparrow y, S by A3,Th37;
A8: ex_inf_of f.:uparrow x, T by A4,A6;
A9: ex_inf_of f.:uparrow y, T by A5,A7;
A10: inf (f.:uparrow x) = f.inf uparrow x by A4,A6;
A11: inf (f.:uparrow y) = f.inf uparrow y by A5,A7;
  assume x <= y;
  then
A12: uparrow y c= uparrow x by Th22;
A13: inf (f.:uparrow x) = f.inf {x} by A10,Th38,YELLOW_0:38;
A14: inf (f.:uparrow y) = f.inf {y} by A11,Th38,YELLOW_0:38;
A15: inf (f.:uparrow x) = f.x by A13,YELLOW_0:39;
  inf (f.:uparrow y) = f.y by A14,YELLOW_0:39;
  hence thesis by A8,A9,A12,A15,RELAT_1:123,YELLOW_0:35;
end;
