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;
reserve X, Y for non empty TopSpace;
reserve f for Function of X,Y;
reserve X,Y,Z for non empty TopSpace;
reserve f for Function of X,Y,
  g for Function of Y,Z;
reserve X, Y for non empty TopSpace,
  X0 for non empty SubSpace of X;
reserve f for Function of X,Y;
reserve f for Function of X,Y,
  X0 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace,
  X0, X1 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;
reserve X0, X1, X2 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;
reserve X for non empty TopSpace,
  H, G for Subset of X;
reserve A for Subset of X;
reserve X0 for non empty SubSpace of X;

theorem
  X0 is open SubSpace of X iff the TopStruct of X = X
  modified_with_respect_to X0
proof
  thus X0 is open SubSpace of X implies the TopStruct of X = X
  modified_with_respect_to X0
  proof
    reconsider A = the carrier of X0 as Subset of X by TSEP_1:1;
    assume X0 is open SubSpace of X;
    then A is open by TSEP_1:def 1;
    then the TopStruct of X = X modified_with_respect_to A by Th95;
    hence thesis by Def10;
  end;
  thus the TopStruct of X = X modified_with_respect_to X0 implies X0 is open
  SubSpace of X
  proof
    assume
A1: the TopStruct of X = X modified_with_respect_to X0;
    now
      let A be Subset of X;
      assume A = the carrier of X0;
      then the TopStruct of X = X modified_with_respect_to A by A1,Def10;
      hence A is open by Th95;
    end;
    hence thesis by TSEP_1:def 1;
  end;
end;
