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).:X = R.:(P.:X)
proof
  for y being object holds y in (P*R).:X iff y in R.:(P.:X)
  proof let y be object;
    thus y in (P*R).:X implies y in R.:(P.:X)
    proof
      assume y in (P*R).:X;
      then consider x such that
A1:   [x,y] in P*R and
A2:   x in X by Def11;
      consider z such that
A3:   [x,z] in P and
A4:   [z,y] in R by A1,Def6;
      z in P.:X by A2,A3,Def11;
      hence thesis by A4,Def11;
    end;
    assume y in R.:(P.:X);
    then consider x such that
A5: [x,y] in R and
A6: x in P.:X by Def11;
    consider z such that
A7: [z,x] in P and
A8: z in X by A6,Def11;
    [z,y] in P*R by A5,A7,Def6;
    hence thesis by A8,Def11;
  end;
  hence thesis by TARSKI:2;
end;
