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;
reserve X1, X2 for non empty SubSpace of X;

theorem
  for Y being non empty SubSpace of X holds X1,X2 are_weakly_separated
  implies X1 union Y,X2 union Y are_weakly_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;
  assume X1,X2 are_weakly_separated;
  then
A1: A1,A2 are_weakly_separated;
  now
    let D1, D2 be Subset of X;
    assume D1 = the carrier of X1 union Y & D2 = the carrier of X2 union Y;
    then A1 \/ C = D1 & A2 \/ C = D2 by Def2;
    hence D1,D2 are_weakly_separated by A1,Th50;
  end;
  hence thesis;
end;
