
theorem Th1:
  for f, g being Function, a being set st f is one-to-one & g is
one-to-one & dom f /\ dom g = { a } & rng f /\ rng g = { f.a } holds f +* g is
  one-to-one
proof
  let f, g be Function, a be set;
  assume that
A1: f is one-to-one and
A2: g is one-to-one and
A3: dom f /\ dom g = { a } and
A4: rng f /\ rng g = { f.a };
  for x1,x2 being set st x1 in dom g & x2 in dom f \ dom g holds g.x1 <> f .x2
  proof
    { a } c= dom g by A3,XBOOLE_1:17;
    then
A5: a in dom g by ZFMISC_1:31;
    { a } c= dom f by A3,XBOOLE_1:17;
    then
A6: a in dom f by ZFMISC_1:31;
    let x1,x2 be set;
    assume that
A7: x1 in dom g and
A8: x2 in dom f \ dom g;
A9: f.x2 in rng f by A8,FUNCT_1:3;
    assume
A10: g.x1 = f.x2;
    g.x1 in rng g by A7,FUNCT_1:3;
    then f.x2 in rng f /\ rng g by A9,A10,XBOOLE_0:def 4;
    then f.x2 = f.a by A4,TARSKI:def 1;
    then x2 = a by A1,A8,A6,FUNCT_1:def 4;
    then dom g meets (dom f \ dom g) by A8,A5,XBOOLE_0:3;
    hence thesis by XBOOLE_1:79;
  end;
  hence thesis by A1,A2,TOPMETR2:1;
end;
