reserve X for TopSpace;

theorem Th5:
  for X being TopStruct for X1, X2 being SubSpace of X st the
carrier of X1 = the carrier of X2 holds the TopStruct of X1 = the TopStruct of
  X2
proof
  let X be TopStruct;
  let X1, X2 be SubSpace of X;
  reconsider S1 = the TopStruct of X1, S2 = the TopStruct of X2 as strict
  SubSpace of X by Lm3;
  set A1 = the carrier of X1, A2 = the carrier of X2;
  assume
A1: A1 = A2;
A2: A1 = [#]S1;
A3: A2 = [#]S2;
  reconsider A1 as Subset of X by BORSUK_1:1;
  S1 = X|(A1) by A2,PRE_TOPC:def 5;
  hence thesis by A1,A3,PRE_TOPC:def 5;
end;
