let T1, T2 be TopSpace; :: thesis: ( the carrier of T1 = the carrier of T2 & ( for A1 being Subset of T1
for A2 being Subset of T2 st A1 = A2 holds
Int A1 = Int A2 ) implies the topology of T1 = the topology of T2 )
assume that
A1: the carrier of T1 = the carrier of T2 and
A2: for A1 being Subset of T1
for A2 being Subset of T2 st A1 = A2 holds
Int A1 = Int A2 ; :: thesis: the topology of T1 = the topology of T2
now :: thesis: for A1 being Subset of T1
for A2 being Subset of T2 st A1 = A2 holds
Cl A1 = Cl A2
let A1 be Subset of T1; :: thesis: for A2 being Subset of T2 st A1 = A2 holds
Cl A1 = Cl A2
let A2 be Subset of T2; :: thesis: ( A1 = A2 implies Cl A1 = Cl A2 )
assume A1 = A2 ; :: thesis: Cl A1 = Cl A2
then Int (A1 `) = Int (A2 `) by A1, A2;
hence Cl A1 = (Int (A2 `)) ` by A1, TDLAT_3:1
.= Cl A2 by TDLAT_3:1 ;
:: thesis: verum
end;
hence the topology of T1 = the topology of T2 by A1, Th8; :: thesis: verum