theorem Th58:
  r > 0 implies ].x - r , x + r.[ is a_neighborhood of x
  proof
    assume
A1: r > 0;
    set S  = ]. x - r, x + r.[,
        BA = {].a,b.[ where a,b is Real: a < b};
    now
      x < x + r & x - r < x by A1,XREAL_1:29,44;
      then x - r < x + r by XXREAL_0:2;
      hence S in BA;
      thus BA c= Family_open_set(FMT_R^1) by Th57,FINTOPO7:def 14;
    end;
    then S in Family_open_set(FMT_R^1);
    then S in the set of all O where O is open Subset of FMT_R^1
      by FINTOPO7:def 11;
    then ex O be open Subset of FMT_R^1 st S = O;
    then S in U_FMT x by A1,TOPREAL6:15,FINTOPO7:def 1;
    hence thesis by FINTOPO7:def 5;
  end;
