reserve S for non void non empty ManySortedSign,
  U1, U2, U3 for non-empty MSAlgebra over S,
  I for set,
  A for ManySortedSet of I,
  B, C for non-empty ManySortedSet of I;

theorem
  for A, B be ManySortedSet of I for M be ManySortedSubset of A for F be
  ManySortedFunction of A, B holds F.:.:M c= F.:.:A
proof
  let A, B be ManySortedSet of I, M be ManySortedSubset of A, F be
  ManySortedFunction of A, B;
  let i be object such that
A1: i in I;
  reconsider f = F.i as Function of A.i, B.i by A1,PBOOLE:def 15;
A2: (F.:.:M).i = f.:(M.i) by A1,PBOOLE:def 20;
  M c= A by PBOOLE:def 18;
  then M.i c= A.i by A1;
  then
A3: f.:(M.i) c= f.:(A.i) by RELAT_1:123;
  let x be object;
  assume x in (F.:.:M).i;
  then x in f.:(A.i) by A2,A3;
  hence thesis by A1,PBOOLE:def 20;
end;
