reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X for TopSpace;
reserve A1, A2 for Subset of X;
reserve A1,A2 for Subset of X;
reserve X for TopSpace,
  A1, A2 for Subset of X;
reserve X for non empty TopSpace,
  A1, A2 for Subset of X;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;

theorem
  for Y being non empty SubSpace of X holds X1,Y are_separated & X2,Y
  are_separated iff X1 union X2,Y are_separated
proof
  reconsider A2 = the carrier of X2 as Subset of X by Th1;
  reconsider A1 = the carrier of X1 as Subset of X by Th1;
  let Y be non empty SubSpace of X;
  reconsider C = the carrier of Y as Subset of X by Th1;
A1: Y is SubSpace of Y by Th2;
  thus X1,Y are_separated & X2,Y are_separated implies X1 union X2,Y
  are_separated
  proof
    assume X1,Y are_separated & X2,Y are_separated;
    then
A2: A1,C are_separated & A2,C are_separated;
    now
      let D, C be Subset of X;
      assume that
A3:   D = the carrier of X1 union X2 and
A4:   C = the carrier of Y;
      A1 \/ A2 = D by A3,Def2;
      hence D,C are_separated by A2,A4,Th41;
    end;
    hence thesis;
  end;
  assume
A5: X1 union X2,Y are_separated;
  X1 is SubSpace of X1 union X2 & X2 is SubSpace of X1 union X2 by Th22;
  hence thesis by A5,A1,Th71;
end;
