theorem Th55:
  A is open iff for x being Real st x in A
  ex r st r > 0 & ].x - r, x + r.[ c= A
  proof
    A is Subset of NTop2Top Top2NTop R^1 by FINTOPO7:def 16;
    then reconsider A9 = A as Subset of R^1 by FINTOPO7:24;
    hereby
      assume A is open;
      then A is open Subset of NTop2Top FMT_R^1 &
        NTop2Top FMT_R^1 = R^1 by FINTOPO7:24,Lm9;
      hence for x being Real st x in A
      ex r being Real st r > 0 & ].x - r, x + r.[ c= A by FRECHET:8;
    end;
    assume for x being Real st x in A
      ex r being Real st r > 0 & ].x - r, x + r.[ c= A;
    then A9 is open by FRECHET:8;
    hence A is open by Lm1;
  end;
