reserve T   for TopSpace,
        A,B for Subset of T;
reserve NT,NTX,NTY for NTopSpace,
        A,B        for Subset of NT,
        O          for open Subset of NT,
        a          for Point of NT,
        XA         for Subset of NTX,
        YB         for Subset of NTY,
        x          for Point of NTX,
        y          for Point of NTY,
        f          for Function of NTX,NTY,
        fc         for continuous Function of NTX,NTY;
reserve NT for T_2 NTopSpace;
reserve T   for non empty TopSpace,
        A,B for Subset of T,
        F   for closed Subset of T,
        O   for open Subset of T;
reserve T   for non empty strict TopSpace,
        A,B for Subset of T,
        x   for Point of T;
reserve A for Subset of FMT_R^1,
        x for Point of FMT_R^1,
        y for Point of RealSpace,
        z for Point of TopSpaceMetr RealSpace,
        r for Real;

theorem Th57:
  {].a,b.[ where a,b is Real: a < b} is Basis of FMT_R^1
  proof
    set BA = {].a,b.[ where a,b is Real: a < b};
    reconsider BBA = BA as open quasi_basis Subset-Family of R^1
      by TOPGEN_5:72;
A1: the topology of R^1 c= UniCl BBA by CANTOR_1:def 2;
A2: the carrier of FMT_R^1 = REAL by TOPMETR:17,FINTOPO7:def 15;
    BA c= bool the carrier of FMT_R^1
    proof
      let x be object;
      assume x in BA;
      then ex a,b be Real st x = ].a,b.[ & a < b;
      then reconsider x as Subset of REAL;
      x in bool the carrier of FMT_R^1 by A2;
      hence thesis;
    end;
    then reconsider BA as Subset-Family of FMT_R^1;
    BA c= Family_open_set(FMT_R^1)
    proof
      let x be object;
      assume
A3:   x in BA;
      then reconsider x as Subset of FMT_R^1;
      consider a,b be Real such that
A4:   x = ].a,b.[ and a < b by A3;
      now
        let y be Real;
        assume
A5:     y in x;
        reconsider z = x as Subset of R^1 by FINTOPO7:def 15;
        z is open by A4,JORDAN6:35;
        hence ex r being Real st r > 0 & ].y - r, y + r.[ c= x by A5,FRECHET:8;
      end;
      then x is open by Th55;
      then x in {O where O is open Subset of FMT_R^1: not contradiction};
      hence thesis by FINTOPO7:def 11;
    end;
    then reconsider BA as open Subset-Family of FMT_R^1 by FINTOPO7:def 14;
    now
      let x be object;
      assume x in Family_open_set(FMT_R^1);
      then x in {O where O is open Subset of FMT_R^1: not contradiction}
        by FINTOPO7:def 11;
      then consider O be open Subset of FMT_R^1 such that
A6:   x = O;
      NTop2Top FMT_R^1 = R^1 by FINTOPO7:24;
      then O is open Subset of R^1 by Lm9;
      then O in the topology of R^1 by PRE_TOPC:def 2;
      then consider Y be Subset-Family of R^1 such that
A7:   Y c= BBA and
A8:   O = union Y by A1,CANTOR_1:def 1;
      reconsider Y as Subset-Family of FMT_R^1 by FINTOPO7:def 15;
      Y c= BA by A7;
      hence x in UniCl BA by A8,A6,CANTOR_1:def 1;
    end;
    then Family_open_set(FMT_R^1) c= UniCl BA;
    hence thesis by FINTOPO7:def 13;
  end;
