reserve i, x, I for set,
  A, M for ManySortedSet of I,
  f for Function,
  F for ManySortedFunction of I;

theorem
  F is "1-1" implies for A, B being ManySortedSet of I st A in doms F &
  B in doms F & F..A = F..B holds A = B
proof
  assume
A1: F is "1-1";
  let A, B be ManySortedSet of I such that
A2: A in doms F & B in doms F and
A3: F..A = F..B;
  now
    let i be object such that
A4: i in I;
    reconsider f = F.i as Function;
A5: f is one-to-one by A1,A4,MSUALG_3:1;
    dom f = (doms F).i by A4,MSSUBFAM:14;
    then
A6: A.i in dom f & B.i in dom f by A2,A4;
    f.(A.i) = (F..A).i by A4,PRALG_1:def 20
      .= f.(B.i) by A3,A4,PRALG_1:def 20;
    hence A.i = B.i by A5,A6;
  end;
  hence thesis;
end;
