reserve x, y, z for set,
  T for TopStruct,
  A for SubSpace of T,
  P, Q for Subset of T;

theorem Th3:
  ( P = {} implies (P is compact iff T|P is compact) ) & ( T is
  TopSpace-like & P <> {} implies (P is compact iff T|P is compact) )
proof
A1: [#](T|P) = P by PRE_TOPC:def 5;
  thus P = {} implies (P is compact iff T|P is compact);
  assume that
A2: T is TopSpace-like and
A3: P <> {};
  reconsider T9 = T as non empty TopSpace by A2,A3;
  reconsider P9 = P as non empty Subset of T9 by A3;
A4: [#](T9|P9) is compact iff T9|P9 is compact;
  hence P is compact implies T|P is compact by A1,Th2;
  assume T|P is compact;
  then for Q being Subset of T|P st Q=P holds Q is compact by A4,PRE_TOPC:def 5
;
  hence thesis by A1,Th2;
end;
