
theorem
  for S, T being non empty TopSpace st the TopStruct of S = the
  TopStruct of T & S is locally-compact holds T is locally-compact
proof
  let S, T be non empty TopSpace such that
A1: the TopStruct of S = the TopStruct of T and
A2: for x being Point of S, X being Subset of S st x in X & X is open ex
  Y being Subset of S st x in Int Y & Y c= X & Y is compact;
  let x be Point of T, X be Subset of T such that
A3: x in X & X is open;
  reconsider A = X as Subset of S by A1;
  consider B being Subset of S such that
A4: x in Int B & B c= A & B is compact by A1,A2,A3,TOPS_3:76;
  reconsider Y = B as Subset of T by A1;
  take Y;
  thus thesis by A1,A4,Th25,TOPS_3:77;
end;
