reserve a,b,c for set;

theorem
  for x,y being Real holds ].x,y.[ is open Subset of Sorgenfrey-line
proof
  let x,y be Real;
  reconsider V = ].x,y.[ as Subset of Sorgenfrey-line by Def2;
  now
    let p be Point of Sorgenfrey-line;
    reconsider a = p as Element of REAL by Def2;
    assume
A1: p in V;
    then a < y by XXREAL_1:4;
    then consider q being Rational such that
A2: a < q and
A3: q < y by RAT_1:7;
    reconsider U = [.a,q.[ as Subset of Sorgenfrey-line by Def2;
    take U;
    thus U in BB by A2,Lm5;
    thus p in U by A2,XXREAL_1:3;
    x < a by A1,XXREAL_1:4;
    hence U c= V by A3,XXREAL_1:48;
  end;
  hence thesis by Lm6,YELLOW_9:31;
end;
