let x, y be Real; :: thesis: ].x,y.[ is open Subset of Sorgenfrey-line
reconsider V = ].x,y.[ as Subset of Sorgenfrey-line by Def2;

now :: thesis: for p being Point of Sorgenfrey-line st p in V holds
ex U being Subset of Sorgenfrey-line st
( U in BB & p in U & U c= V )

let p be Point of Sorgenfrey-line; :: thesis: ( p in V implies ex U being Subset of Sorgenfrey-line st
( U in BB & p in U & U c= V ) )
reconsider a = p as Element of REAL by Def2;
assume A1: p in V ; :: thesis: ex U being Subset of Sorgenfrey-line st
( U in BB & p in U & U c= 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 = U; :: thesis: ( U in BB & p in U & U c= V )
thus U in BB by A2, Lm5; :: thesis: ( p in U & U c= V )
thus p in U by A2, XXREAL_1:3; :: thesis: U c= V
x < a by A1, XXREAL_1:4;
hence U c= V by A3, XXREAL_1:48; :: thesis: verum

end;

hence ].x,y.[ is open Subset of Sorgenfrey-line by Lm6, YELLOW_9:31; :: thesis: verum