
theorem
  for P being mutually-disjoint with_non-empty_elements non empty set
  for f being Choice_Function of P holds f is one-to-one
proof
  let P be mutually-disjoint with_non-empty_elements non empty set;
  let f be Choice_Function of P;
  let x1,x2 be object;
  assume that
A1: x1 in dom f and
A2: x2 in dom f and
A3: f.x1 = f.x2;
   reconsider x1,x2 as set by TARSKI:1;
A4: not {} in P;
  then
A5: f.x1 in x1 by A1,ORDERS_1:89;
  f.x1 in x2 by A2,A3,A4,ORDERS_1:89;
  then x1 meets x2 by A5,XBOOLE_0:3;
  hence thesis by A1,A2,TAXONOM2:def 5;
end;
