reserve p,q,x,x1,x2,y,y1,y2,z,z1,z2 for set;
reserve A,B,V,X,X1,X2,Y,Y1,Y2,Z for set;
reserve C,C1,C2,D,D1,D2 for non empty set;

theorem
  [:id X, id Y:] = id [:X,Y:]
proof
  rng pr1(X,Y) c= X by Th43;
  then
A1: (id X)*pr1(X,Y) = pr1(X,Y) by RELAT_1:53;
  rng pr2(X,Y) c= Y by Th45;
  then
A2: (id Y)*pr2(X,Y) = pr2(X,Y) by RELAT_1:53;
  dom id X = X & dom id Y = Y;
  hence [:id X, id Y:] = <:pr1(X,Y),pr2(X,Y):> by A1,A2,Th66
    .= id [:X,Y:] by Th53;
end;
