reserve a,b,c for set;

theorem Th6:
  for T1,T2 being TopSpace st for A being set holds A is closed
  Subset of T1 iff A is closed 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 closed Subset of T1 iff A is closed
  Subset of T2;
A2: the topology of T2 c= the topology of T1 by A1,Lm2;
  the topology of T1 c= the topology of T2 by A1,Lm2;
  then the topology of T1 = the topology of T2 by A2;
  hence thesis by A1,Lm2;
end;
