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
  X1 union X2 is closed SubSpace of X & X1,X2 are_separated implies X1
  is closed SubSpace of X
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;
  assume
A1: X1 union X2 is closed SubSpace of X;
  assume X1,X2 are_separated;
  then
A2: A1,A2 are_separated;
  A1 \/ A2 = the carrier of X1 union X2 by Def2;
  then A1 \/ A2 is closed by A1,Th11;
  then A1 is closed by A2,Th35;
  hence thesis by Th11;
end;
