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
  for T being non empty TopSpace, X being set holds
  X is compact Subset of T iff X is compact Subset of the TopStruct of T
proof
  let T be non empty TopSpace, X being set;
  thus X is compact Subset of T implies X is compact Subset of the TopStruct
  of T
  proof
    assume
A1: X is compact Subset of T;
    then reconsider Z=X as Subset of the TopStruct of T;
    Z is compact by Th22,A1,Def4;
    hence thesis;
  end;
  assume
A2: X is compact Subset of the TopStruct of T;
  then reconsider Z = X as Subset of T;
  Z is compact by Th22,A2,Def4;
  hence thesis;
end;
