reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem
  (P*R)"Y = P"(R"Y)
proof
  for x being object holds x in (P*R)"Y iff x in P"(R"Y)
  proof let x be object;
    thus x in (P*R)"Y implies x in P"(R"Y)
    proof
      assume x in (P*R)"Y;
      then consider y such that
A1:   [x,y] in P*R and
A2:   y in Y by Def12;
      consider z such that
A3:   [x,z] in P and
A4:   [z,y] in R by A1,Def6;
      z in R"Y by A2,A4,Def12;
      hence thesis by A3,Def12;
    end;
    assume x in P"(R"Y);
    then consider y such that
A5: [x,y] in P and
A6: y in R"Y by Def12;
    consider z such that
A7: [y,z] in R and
A8: z in Y by A6,Def12;
    [x,z] in P*R by A5,A7,Def6;
    hence thesis by A8,Def12;
  end;
  hence thesis by TARSKI:2;
end;
