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

now

let p be Point of Sorgenfrey-line ; :: thesis: ( p in [.x,y.[ 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 p in [.x,y.[ ; :: thesis: ex U being Subset of Sorgenfrey-line st
( U in BB & p in U & U c= V )
then A1: ( x <= a & a < y ) by XXREAL_1:3;
then consider q being rational number such that
A2: ( a < q & q < y ) by RAT_1:22;
reconsider U = [.x,q.[ as Subset of Sorgenfrey-line by Def2;
take U = U; :: thesis: ( U in BB & p in U & U c= V )
( x < q & x is Element of REAL & q is Element of REAL ) by A1, A2, XREAL_0:def 1, XXREAL_0:2;
hence U in BB by Lm5; :: thesis: ( p in U & U c= V )
thus p in U by A1, A2, XXREAL_1:3; :: thesis: U c= V
thus U c= V by A2, XXREAL_1:38; :: thesis: verum

end;

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