reserve a,b,c for set;

theorem Th10:
  for T1,T2 being TopSpace st the carrier of T1 = the carrier of
T2 & for A1 being Subset of T1, A2 being Subset of T2 st A1 = A2 holds Int A1 =
  Int A2 holds the topology of T1 = the topology of T2
proof
  let T1,T2 be TopSpace such that
A1: the carrier of T1 = the carrier of T2 and
A2: for A1 being Subset of T1, A2 being Subset of T2 st A1 = A2 holds
  Int A1 = Int A2;
  now
    let A1 be Subset of T1, A2 be Subset of T2;
    assume A1 = 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;
  end;
  hence thesis by A1,Th8;
end;
