reserve I for set;

theorem Th5:
  for A, B being ManySortedSet of I, F being ManySortedFunction of
A, B for C, E being ManySortedSubset of A, D being ManySortedSubset of C st E =
  D holds (F || C) || D = F || E
proof
  let A, B be ManySortedSet of I, F be ManySortedFunction of A, B, C, E be
  ManySortedSubset of A, D be ManySortedSubset of C such that
A1: E = D;
  now
    let i be object such that
A2: i in I;
    D c= C by PBOOLE:def 18;
    then
A4: D.i c= C.i by A2;
    (F||C).i is Function of C.i, B.i by A2,PBOOLE:def 15;
    then reconsider fc = (F.i) | (C.i) as Function of C.i, B.i by A2,
MSAFREE:def 1;
    thus ((F || C) || D).i = (F || C).i | (D.i) by A2,MSAFREE:def 1
      .= fc | (D.i) by A2,MSAFREE:def 1
      .= F.i | (D.i) by A4,RELAT_1:74
      .= (F || E).i by A1,A2,MSAFREE:def 1;
  end;
  hence thesis;
end;
