reserve x,y for Real,
  u,v,w for set,
  r for positive Real;

theorem Th72:
  {].a,b.[ where a,b is Real: a < b} is Basis of R^1
proof
  set X = {].a,b.[ where a,b is Real: a < b};
  X c= bool REAL
  proof
    let u be object;
    assume u in X;
    then ex a,b being Real st u = ].a,b.[ & a < b;
    hence thesis;
  end;
  then reconsider X as Subset-Family of R^1 by TOPMETR:17;
A1: now
    let U be Subset of R^1 such that
A2: U is open;
    let x be Point of R^1 such that
A3: x in U;
    reconsider a = x as Real;
    consider r being Real such that
A4: 0 < r and
A5: ].a-r,a+r.[ c= U by A2,A3,FRECHET:8;
    reconsider V = ].a-r,a+r.[ as Subset of R^1 by TOPMETR:17;
    take V;
A6: a < a+r by A4,XREAL_1:29;
A7: a-r < a by A4,XREAL_1:44;
    then a-r < a+r by A6,XXREAL_0:2;
    hence V in X;
    thus x in V by A7,A6,XXREAL_1:4;
    thus V c= U by A5;
  end;
  X c= the topology of R^1
  proof
    let u be object;
    assume u in X;
    then ex a, b being Real st u = ].a,b.[ & a < b;
    then u is open Subset of R^1 by JORDAN6:35,TOPMETR:17;
    hence thesis by PRE_TOPC:def 2;
  end;
  hence thesis by A1,YELLOW_9:32;
end;
