let x be Real; :: thesis: left_open_halfline x is open Subset of Sorgenfrey-line

reconsider V = left_open_halfline x as Subset of Sorgenfrey-line by Def2;

reconsider V = left_open_halfline x 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
left_open_halfline x 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 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 = U; :: thesis: ( U in BB & p in U & U c= V )

thus U in BB by A3, Lm5; :: thesis: ( p in U & U c= V )

thus p in U by A3, XXREAL_1:3; :: thesis: U c= V

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; :: 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 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 = U; :: thesis: ( U in BB & p in U & U c= V )

thus U in BB by A3, Lm5; :: thesis: ( p in U & U c= V )

thus p in U by A3, XXREAL_1:3; :: thesis: U c= V

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; :: thesis: verum