reserve U1,U2,U3 for Universal_Algebra,
  m,n for Nat,
  a for set,
  A for non empty set,
  h for Function of U1,U2;

theorem Th8:
  for I be set,I0 be Subset of I, A,B be ManySortedSet of I, F be
ManySortedFunction of A,B for A0,B0 be ManySortedSet of I0 st A0 = A | I0 & B0
  = B | I0 holds F|I0 is ManySortedFunction of A0,B0
proof
  let I be set, I0 be Subset of I, A,B be ManySortedSet of I, F be
  ManySortedFunction of A,B;
  let A0,B0 be ManySortedSet of I0 such that
A1: A0 = A | I0 and
A2: B0 = B | I0;
  reconsider G = F|I0 as ManySortedFunction of I0;
A3: dom A0 = I0 & dom (F|I0) = I0 by PARTFUN1:def 2;
A4: dom B0 = I0 by PARTFUN1:def 2;
  now
    let i be object;
    assume
A5: i in I0;
    then
A6: B.i = B0.i by A2,A4,FUNCT_1:47;
    G.i = F.i & A.i = A0.i by A1,A3,A5,FUNCT_1:47;
    hence G.i is Function of A0.i,B0.i by A5,A6,PBOOLE:def 15;
  end;
  hence thesis by PBOOLE:def 15;
end;
