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;

theorem Th10:
  GX is connected iff
  for A, B being Subset of GX st [#]GX = A \/ B & A <> {}GX & B <> {}GX &
  A is closed & B is closed holds A meets B
proof
A1: now
    assume not GX is connected;
    then consider P, Q being Subset of GX such that
A2: [#]GX = P \/ Q and
A3: P,Q are_separated and
A4: P <> {}GX and
A5: Q <> {}GX;
A6: Q is closed by A2,A3,Th4;
    P is closed by A2,A3,Th4;
    hence
    ex A,B being Subset of GX st [#]GX = A \/ B & A <> {}GX & B <> {}GX &
    A is closed & B is closed & A misses B by A2,A3,A4,A5,A6,Th1;
  end;
   (ex A,B being Subset of GX st [#]GX = A \/ B & A <> {}GX & B <> {}GX &
       A is closed & B is closed & A misses B)
    implies not GX is connected by Th2;
  hence thesis by A1;
end;
