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 p being Point of GX st p in Component_of x holds
  Component_of p = Component_of x
proof
  set A = Component_of x;
A1: A is a_component by Th40;
  given p being Point of GX such that
A2: p in A and
A3: Component_of p <> A;
  Component_of p is a_component by Th40;
  then (Component_of p) misses A by A3,A1,Th1,Th34;
  then
A4: (Component_of p) /\ A = {}GX;
  p in Component_of p by Th38;
  hence contradiction by A2,A4,XBOOLE_0:def 4;
end;
