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 F be ManySortedFunction of A, B for M1, M2 be ManySortedSubset of
  A st M1 c= M2 holds (F||M2).:.:M1 = F.:.:M1
proof
  let F be ManySortedFunction of A, B, M1, M2 be ManySortedSubset of A such
  that
A1: M1 c= M2;
  now
    let i be object;
    assume
A2: i in I;
    then reconsider f = F.i as Function of A.i, B.i by PBOOLE:def 15;
    reconsider fm = (F||M2).i as Function of M2.i, B.i by A2,PBOOLE:def 15;
A3: M1.i c= M2.i by A1,A2;
    fm = f|(M2.i) by A2,MSAFREE:def 1;
    hence ((F||M2).:.:M1).i = (f|(M2.i)).:(M1.i) by A2,PBOOLE:def 20
      .= f.:(M1.i) by A3,RELAT_1:129
      .= (F.:.:M1).i by A2,PBOOLE:def 20;
  end;
  hence thesis;
end;
