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;
