reserve I, G, H for set, i, x for object,
  A, B, M for ManySortedSet of I,
  sf, sg, sh for Subset-Family of I,
  v, w for Subset of I,
  F for ManySortedFunction of I;

theorem
  A is_transformable_to B & F is ManySortedFunction of A, B implies
  doms F = A & rngs F c= B
proof
  assume that
A1: A is_transformable_to B and
A2: F is ManySortedFunction of A, B;
  now
    let i be object;
    assume
A3: i in I;
    then reconsider f = F.i as Function of A.i, B.i by A2,PBOOLE:def 15;
A4: B.i = {} implies A.i = {} by A1,A3,PZFMISC1:def 3;
    thus (doms F).i = dom f by A3,Th14
      .= A.i by A4,FUNCT_2:def 1;
  end;
  hence doms F = A;
  let i be object;
  assume
A5: i in I;
  then reconsider f = F.i as Function of A.i, B.i by A2,PBOOLE:def 15;
  rng f c= B.i by RELAT_1:def 19;
  hence thesis by A5,Th13;
end;
