reserve r,r1,r2, s,x for Real,
  i for Integer;

theorem Th2:
  for f being Function, X, Y being set st f|X is one-to-one & Y c=
  X holds f|Y is one-to-one
proof
  let f be Function, X, Y be set such that
A1: f|X is one-to-one and
A2: Y c= X;
  let x, y be object;
A3: dom (f|Y) = dom f /\ Y by RELAT_1:61;
  assume
A4: x in dom (f|Y);
  then
A5: x in Y by A3,XBOOLE_0:def 4;
  x in dom f by A3,A4,XBOOLE_0:def 4;
  then x in dom f /\ X by A2,A5,XBOOLE_0:def 4;
  then
A6: x in dom (f|X) by RELAT_1:61;
  assume
A7: y in dom (f|Y);
  then
A8: y in Y by A3,XBOOLE_0:def 4;
  y in dom f by A3,A7,XBOOLE_0:def 4;
  then y in dom f /\ X by A2,A8,XBOOLE_0:def 4;
  then
A9: y in dom (f|X) by RELAT_1:61;
  assume
A10: (f|Y).x = (f|Y).y;
  (f|X).x = f.x by A2,A5,FUNCT_1:49
    .= (f|Y).x by A5,FUNCT_1:49
    .= f.y by A8,A10,FUNCT_1:49
    .= (f|X).y by A2,A8,FUNCT_1:49;
  hence thesis by A1,A6,A9;
end;
