reserve a,b,c for set;

theorem
  for x being Real holds right_closed_halfline x is open Subset
  of Sorgenfrey-line
proof
  let x be Real;
  reconsider V = right_closed_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;
    a+0 < a+1 by XREAL_1:6;
    then consider q being Rational such that
A1: a < q and
    q < a+1 by RAT_1:7;
    a in [.a,q.] by A1,XXREAL_1:1;
    then
A2: {a} c= [.a,q.] by ZFMISC_1:31;
    reconsider U = [.a,q.[ as Subset of Sorgenfrey-line by Def2;
    assume p in V;
    then a >= x by XXREAL_1:236;
    then
A3: [.a,q.] c= V by XXREAL_1:251;
    take U;
    thus U in BB by A1,Lm5;
    thus p in U by A1,XXREAL_1:3;
A4: ].a,q.[ c= [.a, q .] by XXREAL_1:37;
    U = {a}\/].a,q.[ by A1,XXREAL_1:131;
    then U c= [.a,q.] by A2,A4,XBOOLE_1:8;
    hence U c= V by A3;
  end;
  hence thesis by Lm6,YELLOW_9:31;
end;
