reserve GX for TopSpace;
reserve A, B, C for Subset of GX;
reserve TS for TopStruct;
reserve K, K1, L, L1 for Subset of TS;
reserve GX for non empty TopSpace;
reserve A, C for Subset of GX;
reserve x for Point of GX;

theorem
  for G being non empty TopSpace, P being Subset of G,A being
Subset of G, Q being Subset of G|A st P=Q & P is connected holds Q is connected
proof
  let G be non empty TopSpace,P be Subset of G, A be Subset of G, Q be Subset
  of G|A;
  assume that
A1: P=Q and
A2: P is connected;
A3: G|P is connected by A2;
  Q c= the carrier of G|A;
  then Q c= A by PRE_TOPC:8;
  then G|P=(G|A) |Q by A1,PRE_TOPC:7;
  hence thesis by A3;
end;
