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;
