reserve a,b,c for set;

theorem
  for x being Real holds left_open_halfline x is open Subset of
  Sorgenfrey-line
proof
  let x be Real;
  reconsider V = left_open_halfline x 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
A2: {a} c= V by ZFMISC_1:31;
    a < x by A1,XXREAL_1:233;
    then consider q being Rational such that
A3: a < q and
A4: q < x by RAT_1:7;
    reconsider U = [.a,q.[ as Subset of Sorgenfrey-line by Def2;
    take U;
    thus U in BB by A3,Lm5;
    thus p in U by A3,XXREAL_1:3;
A5: ].a,q.[ c= V by A4,XXREAL_1:263;
    U = {a}\/].a,q.[ by A3,XXREAL_1:131;
    hence U c= V by A2,A5,XBOOLE_1:8;
  end;
  hence thesis by Lm6,YELLOW_9:31;
end;
