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,Y are_weakly_separated &
  X2,Y are_weakly_separated implies X1 union X2,Y are_weakly_separated) & (Y,X1
  are_weakly_separated & Y,X2 are_weakly_separated implies Y,X1 union X2
  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;
  thus X1,Y are_weakly_separated & X2,Y are_weakly_separated implies X1
  union X2,Y are_weakly_separated
  proof
    assume X1,Y are_weakly_separated & X2,Y are_weakly_separated;
    then
A1: A1,C are_weakly_separated & A2,C are_weakly_separated;
    now
      let D, C be Subset of X;
      assume that
A2:   D = the carrier of X1 union X2 and
A3:   C = the carrier of Y;
      A1 \/ A2 = D by A2,Def2;
      hence D,C are_weakly_separated by A1,A3,Th53;
    end;
    hence thesis;
  end;
  hence thesis;
end;
