
theorem Th79:
  for f being Function, x, z being object
  st x in dom f & z in rng renameElementsDistinctlyFunc(f,x)
  ex y being object st y in f.x & z = [f,x,y]
proof
  let f be Function, x, z be object;
  assume A1: x in dom f & z in rng renameElementsDistinctlyFunc(f,x);
  then consider y being object such that
    A2: y in dom renameElementsDistinctlyFunc(f,x) and
    A3: renameElementsDistinctlyFunc(f,x).y = z by FUNCT_1:def 3;
  take y;
  thus y in f.x by A2;
  hence thesis by A1, A3, Th78;
end;
