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
  A is a_component & x in A implies A = Component_of x
proof
  assume that
A1: A is a_component and
A2: x in A;
  x in Component_of x by Th38;
  then A /\ (Component_of x) <> {} by A2,XBOOLE_0:def 4;
  then
A3: A meets (Component_of x);
A4: Component_of x is a_component by Th40;
  assume A <> Component_of x;
  hence contradiction by A1,A3,A4,Th1,Th34;
end;
