
theorem Th2:
  for X, Y being non empty set, A, B being Subset of X, f being
  Function of X,Y st f.:A misses f.:B holds A misses B
proof
  let X, Y be non empty set;
  let A, B be Subset of X;
  let f be Function of X,Y such that
A1: f.:A /\ f.:B = {};
  assume A /\ B <> {};
  then consider x being Element of X such that
A2: x in A /\ B by SUBSET_1:4;
  x in B by A2,XBOOLE_0:def 4;
  then
A3: f.x in f.:B by FUNCT_2:35;
  x in A by A2,XBOOLE_0:def 4;
  then f.x in f.:A by FUNCT_2:35;
  hence contradiction by A1,A3,XBOOLE_0:def 4;
end;
