
theorem
  for F being compatible functional set
  st (for f1 being Function st f1 in F holds f1 is one-to-one &
    for f2 being Function st f2 in F & f1 <> f2 holds rng f1 misses rng f2)
  holds union F is one-to-one
proof
  let F be compatible functional set;
  assume A1: for f1 being Function st f1 in F holds f1 is one-to-one &
    for f2 being Function st f2 in F & f1 <> f2 holds rng f1 misses rng f2;
  now
    let x1, x2 be object;
    assume A2: x1 in dom union F & x2 in dom union F &
      (union F).x1 = (union F).x2;
    then [x1,(union F).x1] in union F by FUNCT_1:1;
    then consider X1 being set such that
      A3: [x1,(union F).x1] in X1 & X1 in F by TARSKI:def 4;
    reconsider f1 = X1 as Function by A3;
    A4: x1 in dom f1 & f1.x1 = (union F).x1 by A3, FUNCT_1:1;
    [x2,(union F).x2] in union F by A2, FUNCT_1:1;
    then consider X2 being set such that
      A5: [x2,(union F).x2] in X2 & X2 in F by TARSKI:def 4;
    reconsider f2 = X2 as Function by A5;
    A6: x2 in dom f2 & f2.x2 = (union F).x2 by A5, FUNCT_1:1;
    then A7: f1.x1 = f2.x2 by A2, A4;
    f1.x1 in rng f1 & f2.x2 in rng f2 by A4, A6, FUNCT_1:3;
    then f1 = f2 by A1, A3, A5, A7, XBOOLE_0:3;
    hence x1 = x2 by A1, A3, A4, A6, A7, FUNCT_1:def 4;
  end;
  hence thesis by FUNCT_1:def 4;
end;
