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 X be ManySortedSubset of A st
  A c= X holds F || X = F
proof
  let F be ManySortedFunction of A, B, X be ManySortedSubset of A such that
A1: A c= X;
  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;
A3: A.i c= X.i by A1,A2;
    thus (F || X).i = (f | (X.i)) by A2,MSAFREE:def 1
      .= F.i by A3,RELSET_1:19;
  end;
  hence thesis;
end;
