reserve a,b,c,d,x,y,z for object, X,Y,Z for set;
reserve R,S,T for Relation;
reserve F,G for Function;

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;
