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
  X = X1 union X2 & X1 misses X2 implies (X1,X2 are_weakly_separated iff
  X1 is closed SubSpace of X & X2 is closed SubSpace of X)
proof
  assume
A1: X = X1 union X2;
  assume
A2: X1 misses X2;
  thus X1,X2 are_weakly_separated implies X1 is closed SubSpace of X & X2 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 X1,X2 are_weakly_separated;
    then X1,X2 are_separated by A2,Th78;
    then
A3: A1,A2 are_separated;
    A1 \/ A2 = [#]X by A1,Def2;
    then A1 is closed & A2 is closed by A3,CONNSP_1:4;
    hence thesis by Th11;
  end;
  thus thesis by Th80;
end;
