
theorem
  for I being set, A being ManySortedSet of I holds (id Union A)-MSF(I,A
  ) = id A
proof
  let I be set, A be ManySortedSet of I;
  set f = id Union A, F = f-MSF(I,A);
  now
    let i be object;
A1: Union A = union rng A by CARD_3:def 4;
    assume
A2: i in I;
    then i in dom A by PARTFUN1:def 2;
    then
A3: A.i in rng A by FUNCT_1:def 3;
    F.i = f|(A.i) & (id A).i = id (A.i) by A2,Def1,MSUALG_3:def 1;
    hence F.i = (id A).i by A3,A1,FUNCT_3:1,ZFMISC_1:74;
  end;
  hence thesis;
end;
