reserve a,b,c for set;

theorem
  for T1,T2 being TopSpace st for A being set holds A is open Subset of
  T1 iff A is open Subset of T2 holds the TopStruct of T1 = the TopStruct of T2
proof
  let T1,T2 be TopSpace such that
A1: for A being set holds A is open Subset of T1 iff A is open Subset of T2;
A2: the topology of T2 c= the topology of T1 by A1,Lm1;
  the topology of T1 c= the topology of T2 by A1,Lm1;
  then the topology of T1 = the topology of T2 by A2;
  hence thesis by A1,Lm1;
end;
