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 Th27:
  [#]GX is connected iff GX is connected
proof
A1: [#]GX = [#](GX|[#]GX) by PRE_TOPC:def 5;
A2: now
    assume
A3: GX is connected;
    now
      let P1,Q1 be Subset of GX|([#]GX) such that
A4:   [#](GX|[#]GX) = P1 \/ Q1 and
A5:   P1,Q1 are_separated;
      reconsider Q = Q1 as Subset of GX by PRE_TOPC:11;
      reconsider P = P1 as Subset of GX by PRE_TOPC:11;
      P,Q are_separated by A5,Th5;
      hence P1 = {}(GX|([#]GX)) or Q1 = {}(GX|([#]GX)) by A1,A3,A4;
    end;
    then GX|([#]GX) is connected;
    hence [#]GX is connected;
  end;
  now
    assume [#]GX is connected;
    then
A6: GX|[#]GX is connected;
    now
      let P1,Q1 be Subset of GX such that
A7:   [#]GX = P1 \/ Q1 and
A8:   P1,Q1 are_separated;
      reconsider Q = Q1 as Subset of GX|([#]GX) by A1;
      reconsider P = P1 as Subset of (GX|([#]GX)) by A1;
      P,Q are_separated by A1,A7,A8,Th6;
      hence P1 = {}GX or Q1 = {}GX by A1,A6,A7;
    end;
    hence GX is connected;
  end;
  hence thesis by A2;
end;
