reserve a,x,y for object, A,B for set,
  l,m,n for Nat;

theorem
  for X,Y be non empty set, f being Function of X,Y st f is one-to-one
for y being Element of Y, A being Subset of X, B being Subset of Y st y in f.:A
  \ B holds f".y in A \ f"B
proof
  let X,Y be non empty set, f be Function of X,Y such that
A1: f is one-to-one;
  let y be Element of Y, A be Subset of X, B be Subset of Y;
  assume
A2: y in f.:A \ B;
  then
A3: y in f.:A by XBOOLE_0:def 5;
A4: f.:A c= rng f by RELAT_1:111;
  then f".y in dom f by A1,A3,FUNCT_1:32;
  then reconsider x = f".y as Element of X;
  y = f.x by A1,A3,A4,FUNCT_1:35;
  hence thesis by A1,A2,Th4;
end;
