reserve A, B for non empty set,
  A1, A2, A3 for non empty Subset of A;
reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;

theorem Th41:
  for X0 being open non empty SubSpace of X holds X1 is SubSpace
  of X0 & X0 misses X2 implies X1 is open SubSpace of X1 union X2 & X1 is open
  SubSpace of X2 union X1
proof
A1: X1 is SubSpace of X1 union X2 by TSEP_1:22;
  reconsider S = the TopStruct of X1 as SubSpace of X by Th6;
  let X0 be open non empty SubSpace of X;
  assume
A2: X1 is SubSpace of X0;
  assume X0 misses X2;
  then
A3: X0 meet (X1 union X2) = the TopStruct of X1 by A2,Th28;
  X0 meets X1 by A2,Th17;
  then X0 meets X1 union X2 by A1,Th18;
  then S is open SubSpace of X1 union X2 by A3,Th39;
  hence thesis by Th9;
end;
