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 Th20:
  GX is connected & A is connected & [#]GX \ A = B \/ C & B,C
  are_separated implies A \/ B is connected & A \/ C is connected
proof
  assume that
A1: GX is connected and
A2: A is connected and
A3: [#]GX \ A = B \/ C and
A4: B,C are_separated;
  now
    let A,B,C be Subset of GX such that
A5: GX is connected and
A6: A is connected and
A7: [#]GX \ A = B \/ C and
A8: B,C are_separated;
    now
      let P,Q be Subset of GX such that
A9:   A \/ B = P \/ Q and
A10:  P,Q are_separated;
A11:  [#]GX = A \/ (B \/ C) by A7,XBOOLE_1:45
        .= P \/ Q \/ C by A9,XBOOLE_1:4;
A12:  now
        assume A c= Q;
        then P misses A by A10,Th1,Th7;
        then P c= B by A9,XBOOLE_1:7,73;
        then P,C are_separated by A8,Th7;
        then
A13:    P,Q \/ C are_separated by A10,Th8;
        [#]GX = P \/ (Q \/ C) by A11,XBOOLE_1:4;
        hence P = {}GX or Q = {}GX by A5,A13;
      end;
      now
        assume A c= P;
        then Q misses A by A10,Th1,Th7;
        then Q c= B by A9,XBOOLE_1:7,73;
        then Q,C are_separated by A8,Th7;
        then
A14:    Q,P \/ C are_separated by A10,Th8;
        [#]GX = Q \/ (P \/ C) by A11,XBOOLE_1:4;
        hence P = {}GX or Q = {}GX by A5,A14;
      end;
      hence P = {}GX or Q = {}GX by A6,A9,A10,A12,Th16,XBOOLE_1:7;
    end;
    hence A \/ B is connected by Th15;
  end;
  hence thesis by A1,A2,A3,A4;
end;
