reserve f for Function;
reserve p,q for FinSequence;
reserve A,B,C for set,x,x1,x2,y,z for object;
reserve k,l,m,n for Nat;
reserve a for Nat;

theorem
  (for x st x in dom f holds f is_one-to-one_at x) iff f is one-to-one
proof
  thus (for x st x in dom f holds f is_one-to-one_at x) implies f is one-to-one
  proof
    assume
A1: for x st x in dom f holds f is_one-to-one_at x;
    let x1,x2 be object;
    assume that
A2: x1 in dom f and
A3: x2 in dom f & f.x1 = f.x2;
    f is_one-to-one_at x1 by A1,A2;
    hence thesis by A3,Th3;
  end;
  assume
A4: f is one-to-one;
  let x;
  assume
A5: x in dom f;
  then for z holds z in dom f & x <> z implies f.x <> f.z by A4;
  hence thesis by A5,Th3;
end;
