
theorem INV00:
  for X,Y be non empty set, f be Function of X,Y, g be Function of Y,X
  st (for x be Element of X holds g.(f.x) = x) &
  (for y be Element of Y holds f.(g.y) = y)
  holds f is one-to-one & f is onto & g is one-to-one & g is onto &
  g = f" & f = g"
proof
  let X,Y be non empty set, f be Function of X,Y, g be Function of Y,X;
  assume
A1: for x be Element of X holds g.(f.x) = x;
  assume
A2: for y be Element of Y holds f.(g.y) = y;
  now let x be Element of X;
    thus (g*f).x = g.(f.x) by FUNCT_2:15
      .= x by A1;
  end;
  then
P2: g*f = id X by FUNCT_2:124;
  now let y be Element of Y;
    thus (f*g).y = f.(g.y) by FUNCT_2:15
      .= y by A2;
  end;
  then
P4: f*g = id Y by FUNCT_2:124;
  thus
P5: f is one-to-one & f is onto by P2,P4,FUNCT_2:23;
  thus
P6: g is one-to-one & g is onto by P2,P4,FUNCT_2:23;
  rng f = Y by P5,FUNCT_2:def 3;
  hence g = f" by FUNCT_2:30,P2,FUNCT_2:23;
  rng g = X by P6,FUNCT_2:def 3;
  hence f = g" by FUNCT_2:30,P4,FUNCT_2:23;
end;
