theorem
  ex A, B being Subset of R^1 st A is boundary & B is boundary &
  A \/ B is non boundary
proof
  reconsider B = IRRAT as Subset of R^1 by TOPMETR:17;
  reconsider A = RAT as Subset of R^1 by NUMBERS:12,TOPMETR:17;
  take A,B;
  A \/ B = RAT \/ REAL by BORSUK_5:def 1,XBOOLE_1:39
    .= REAL by NUMBERS:12,XBOOLE_1:12;
  hence thesis by Th53,Th54,Th55;
end;
