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 st A is_transformable_to B for F be
ManySortedFunction of A, B for C be ManySortedSet of I st B is ManySortedSubset
  of C holds F is ManySortedFunction of A, C
proof
  let A, B be ManySortedSet of I such that
A1: A is_transformable_to B;
  let F be ManySortedFunction of A, B, C be ManySortedSet of I;
  assume B is ManySortedSubset of C;
  then
A2: B c= C by PBOOLE:def 18;
  let i be object such that
A3: i in I;
A4: B.i = {} implies A.i = {} by A1,A3,PZFMISC1:def 3;
A5: F.i is Function of A.i, B.i by A3,PBOOLE:def 15;
  (B.i) c= (C.i) by A3,A2;
  hence thesis by A4,A5,FUNCT_2:7;
end;
