theorem
  f is one-to-one & y in rng f implies (f").y = f <- y
proof
  assume that
A1: f is one-to-one and
A2: y in rng f;
  consider x being object such that
A3: x in dom f & f.x = y by A2,FUNCT_1:def 3;
  f just_once_values y by A1,A2,Th8;
  then x = f <- y by A3,Def3;
  hence thesis by A1,A3,FUNCT_1:32;
end;
