let T be non empty transitive lower TopRelStr ; :: thesis: for A being Subset of T st A is open holds
A is lower
let A be Subset of T; :: thesis: ( A is open implies A is lower )
assume A1: A in the topology of T ; :: according to PRE_TOPC:def 5 :: thesis: A is lower
reconsider BB = { ((uparrow x) ` ) where x is Element of T : verum } as prebasis of T by Def1;
consider F being Basis of T such that
A2: F c= FinMeetCl BB by CANTOR_1:def 5;
the topology of T c= UniCl F by CANTOR_1:def 2;
then consider Y being Subset-Family of T such that
A3: ( Y c= F & A = union Y ) by A1, CANTOR_1:def 1;
let x, y be Element of T; :: according to WAYBEL_0:def 19 :: thesis: ( not x in A or not y <= x or y in A )
assume x in A ; :: thesis: ( not y <= x or y in A )
then consider Z being set such that
A4: ( x in Z & Z in Y ) by A3, TARSKI:def 4;
Z in F by A3, A4;
then consider X being Subset-Family of T such that
A5: ( X c= BB & X is finite & Z = Intersect X ) by A2, CANTOR_1:def 4;
assume A6: x >= y ; :: thesis: y in A

now

let Q be set ; :: thesis: ( Q in X implies y in Q )
assume A7: Q in X ; :: thesis: y in Q
then Q in BB by A5;
then consider z being Element of T such that
A8: Q = (uparrow z) ` ;
( x in Q & uparrow z misses Q ) by A4, A5, A7, A8, SETFAM_1:58, XBOOLE_1:79;
then ( x in Q & (uparrow z) /\ Q = {} T ) by XBOOLE_0:def 7;
then not x in uparrow z by XBOOLE_0:def 4;
then not x >= z by WAYBEL_0:18;
then not y >= z by A6, ORDERS_2:26;
then not y in uparrow z by WAYBEL_0:18;
hence y in Q by A8, XBOOLE_0:def 5; :: thesis: verum

end;

then y in Z by A5, SETFAM_1:58;
hence y in A by A3, A4, TARSKI:def 4; :: thesis: verum