theorem Th64:
  for X being TopStruct, F being Subset-Family of X holds
  F is open iff F c= the topology of X
  proof
    let X be TopStruct, F be Subset-Family of X;
    thus F is open implies F c= the topology of X
    proof
      assume
A1:   F is open;
      let A be object;
      assume
A2:   A in F;
      then reconsider A as Subset of X;
      A is open by A1,A2;
      hence thesis;
    end;
    assume
A3: F c= the topology of X;
    let A be Subset of X;
    thus thesis by A3;
  end;
