theorem
  [:X,X:] = [:Y,Y:] implies X = Y
proof
  assume
A1: [:X,X:] = [:Y,Y:];
  for x holds x in X iff x in Y
  proof let x;
    x in X iff [x,x] in [:X,X:] by Lm16;
    hence thesis by A1,Lm16;
  end;
  hence thesis by TARSKI:2;
end;
