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 Y1, Y2 being non empty SubSpace of X st Y1 is SubSpace of X2 & Y2
is SubSpace of X1 holds X1,X2 are_weakly_separated implies X1 union Y1,X2 union
  Y2 are_weakly_separated & Y1 union X1,Y2 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 Y1, Y2 be non empty SubSpace of X such that
A1: Y1 is SubSpace of X2 & Y2 is SubSpace of X1;
  reconsider B2 = the carrier of Y2 as Subset of X by Th1;
  reconsider B1 = the carrier of Y1 as Subset of X by Th1;
  assume X1,X2 are_weakly_separated;
  then
A2: A1,A2 are_weakly_separated;
A3: B1 c= A2 & B2 c= A1 by A1,Th4;
A4: now
    let D1, D2 be Subset of X;
    assume D1 = the carrier of X1 union Y1 & D2 = the carrier of X2 union Y2;
    then A1 \/ B1 = D1 & A2 \/ B2 = D2 by Def2;
    hence D1,D2 are_weakly_separated by A3,A2,Th51;
  end;
  hence X1 union Y1,X2 union Y2 are_weakly_separated;
  thus thesis by A4;
end;
