
theorem
  for f being Function, X, x being set st f|X is one-to-one & x in rng (
  f|X) holds (f*(f|X)").x = x
proof
  let f be Function, X, x be set;
  set g = f|X;
  assume that
A1: g is one-to-one and
A2: x in rng g;
  consider a being object such that
A3: a in dom g and
A4: g.a = x by A2,FUNCT_1:def 3;
  dom g = dom f /\ X by RELAT_1:61;
  then
A5: a in X by A3,XBOOLE_0:def 4;
  dom (g") = rng g by A1,FUNCT_1:32;
  hence (f*g").x = f.(g".x) by A2,FUNCT_1:13
    .= f.a by A1,A3,A4,FUNCT_1:32
    .= x by A4,A5,FUNCT_1:49;
end;
