theorem
  (R |_2 Y) |_2 Y = R |_2 Y
proof
  let a,b be object;
  thus [a,b] in (R |_2 Y) |_2 Y implies [a,b] in R |_2 Y by XBOOLE_0:def 4;
  assume
A1: [a,b] in R |_2 Y;
  then [a,b] in [:Y,Y:] by XBOOLE_0:def 4;
  hence thesis by A1,XBOOLE_0:def 4;
end;
