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;

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 )

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