reserve x, y, z for set,
  T for TopStruct,
  A for SubSpace of T,
  P, Q for Subset of T;
reserve TS for TopSpace;
reserve PS, QS for Subset of TS;
reserve S for non empty TopStruct;
reserve f for Function of T,S;
reserve SS for non empty TopSpace;
reserve f for Function of TS,SS;
reserve T, S for non empty TopSpace,
  p for Point of T;

theorem Th22:
  for T being non empty TopSpace, F being set holds F is open
  Subset-Family of T iff F is open Subset-Family of the TopStruct of T
proof
  let T be non empty TopSpace, F be set;
  thus F is open Subset-Family of T implies F is open Subset-Family of the
  TopStruct of T
  proof
    assume
A1: F is open Subset-Family of T;
    then reconsider F as Subset-Family of the TopStruct of T;
    F is open
    by A1,TOPS_2:def 1,PRE_TOPC:30;
    hence thesis;
  end;
  assume
A2: F is open Subset-Family of the TopStruct of T;
  then reconsider F as Subset-Family of T;
  F is open
  by A2,TOPS_2:def 1,PRE_TOPC:30;
  hence thesis;
end;
