
theorem
  for f being Function st f is one-to-one for g being rng-retract of f
  holds g = f"
proof
  let f be Function such that
A1: f is one-to-one;
  let g be rng-retract of f;
A2: rng f = dom g by Def2;
A3: rng g = dom f
  proof
    thus rng g c= dom f by Th23;
    let x be object;
    assume
A4: x in dom f;
    then
A5: f.x in rng f by FUNCT_1:def 3;
    then
A6: g.(f.x) in dom f by Th24;
    f.(g.(f.x)) = f.x by A5,Th24;
    then x = g.(f.x) by A1,A4,A6;
    hence thesis by A2,A5,FUNCT_1:def 3;
  end;
  now
    let x,y be object;
    assume that
A7: x in dom f and
A8: y in dom g;
A9: g.y in rng g by A8,FUNCT_1:def 3;
    f.(g.y) = y by A2,A8,Th24;
    hence f.x = y iff g.y = x by A1,A3,A7,A9;
  end;
  hence thesis by A1,A2,A3,FUNCT_1:38;
end;
