
theorem Th72:
  for S,T being non empty Poset, f being Function of S,T st
  for X being Ideal of S holds f preserves_sup_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 Ideal of S holds f preserves_sup_of X;
  let x,y be Element of S;
A2: ex_sup_of {x}, S by YELLOW_0:38;
A3: ex_sup_of {y}, S by YELLOW_0:38;
A4: f preserves_sup_of downarrow x by A1;
A5: f preserves_sup_of downarrow y by A1;
A6: ex_sup_of downarrow x, S by A2,Th32;
A7: ex_sup_of downarrow y, S by A3,Th32;
A8: ex_sup_of f.:downarrow x, T by A4,A6;
A9: ex_sup_of f.:downarrow y, T by A5,A7;
A10: sup (f.:downarrow x) = f.sup downarrow x by A4,A6;
A11: sup (f.:downarrow y) = f.sup downarrow y by A5,A7;
  assume x <= y;
  then
A12: downarrow x c= downarrow y by Th21;
A13: sup (f.:downarrow x) = f.sup {x} by A10,Th33,YELLOW_0:38;
A14: sup (f.:downarrow y) = f.sup {y} by A11,Th33,YELLOW_0:38;
A15: sup (f.:downarrow x) = f.x by A13,YELLOW_0:39;
  sup (f.:downarrow y) = f.y by A14,YELLOW_0:39;
  hence thesis by A8,A9,A12,A15,RELAT_1:123,YELLOW_0:34;
end;
