reserve I,J for set,i,j,x for object,
  S for non empty ManySortedSign;

theorem
  for I be non empty set, J be set, A,B be ManySortedSet of I,
      f be Function of J,I holds [|A,B|]*f = [|A*f,B*f|]
proof
  let I be non empty set, J be set, A,B be ManySortedSet of I, f be Function
  of J,I;
  for i being object st i in J holds ([|A,B|]*f).i = ([|A*f,B*f|]).i
  proof
A1: dom (B*f) = J by PARTFUN1:def 2;
    let i be object;
A2: dom (A*f) = J by PARTFUN1:def 2;
    assume
A3: i in J; then
A4: f.i in I by FUNCT_2:5;
    dom ([|A,B|]*f) = J by PARTFUN1:def 2;
    hence ([|A,B|]*f).i = [|A,B|].(f.i) by A3,FUNCT_1:12
      .= [:A.(f.i),B.(f.i):] by A4,PBOOLE:def 16
      .= [:(A*f).i,B.(f.i):] by A3,A2,FUNCT_1:12
      .= [:(A*f).i,(B*f).i:] by A3,A1,FUNCT_1:12
      .= [|A*f,B*f|].i by A3,PBOOLE:def 16;
  end;
  hence thesis;
end;
