
theorem Th82:
  for f being Function, x1, x2 being object st x1 <> x2
  holds rng renameElementsDistinctlyFunc(f,x1) misses
    rng renameElementsDistinctlyFunc(f,x2)
proof
  let f be Function, x1, x2 be object;
  assume A1: x1 <> x2;
  per cases;
  suppose A2: x1 in dom f & x2 in dom f;
    rng renameElementsDistinctlyFunc(f,x1) /\
      rng renameElementsDistinctlyFunc(f,x2) = {}
    proof
      assume rng renameElementsDistinctlyFunc(f,x1) /\
        rng renameElementsDistinctlyFunc(f,x2) <> {};
      then consider z being object such that
        A3: z in rng renameElementsDistinctlyFunc(f,x1) /\
          rng renameElementsDistinctlyFunc(f,x2) by XBOOLE_0:def 1;
      A4: z in rng renameElementsDistinctlyFunc(f,x1) &
        z in rng renameElementsDistinctlyFunc(f,x2) by A3, XBOOLE_0:def 4;
      then consider y1 being object such that
        A5: y1 in f.x1 & z = [f,x1,y1] by A2, Th79;
      consider y2 being object such that
        A6: y2 in f.x2 & z = [f,x2,y2] by A2, A4, Th79;
      thus contradiction by A1, A5, A6, XTUPLE_0:3;
    end;
    hence thesis by XBOOLE_0:def 7;
  end;
  suppose not x1 in dom f or not x2 in dom f;
    then f.x1 = {} or f.x2 = {} by FUNCT_1:def 2;
    then renameElementsDistinctlyFunc(f,x1) = {} or
      renameElementsDistinctlyFunc(f,x2) = {};
    then rng renameElementsDistinctlyFunc(f,x1) = {} or
      rng renameElementsDistinctlyFunc(f,x2) = {};
    hence thesis by XBOOLE_1:65;
  end;
end;
