
theorem
  for S, T being non empty RelStr, D being Subset of S,
  f being Function of S, T st ex_inf_of D,S & ex_inf_of f.:D,T or
  S is complete antisymmetric & T is complete antisymmetric holds
  f is monotone implies f.(inf D) <= inf (f.:D)
proof
  let S, T be non empty RelStr;
  let D be Subset of S;
  let f be Function of S, T;
  assume that
A1: ex_inf_of D,S & ex_inf_of f.:D,T or
  S is complete antisymmetric & T is complete antisymmetric;
A2: ex_inf_of D,S by A1,YELLOW_0:17;
A3: ex_inf_of f.:D,T by A1,YELLOW_0:17;
  assume
A4: f is monotone;
  inf D is_<=_than D by A2,YELLOW_0:def 10;
  then f.(inf D) is_<=_than f.:D by A4,YELLOW_2:13;
  hence thesis by A3,YELLOW_0:def 10;
end;
