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;
reserve X, Y for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace;

theorem
  for X1, X2 being non empty SubSpace of X holds X1,X2 are_separated iff
X1 misses X2 & for Y being non empty TopSpace, f1 being continuous Function of
  X1,Y, f2 being continuous Function of X2,Y holds f1 union f2 is continuous
  Function of X1 union X2,Y
proof
  let X1, X2 be non empty SubSpace of X;
  thus X1,X2 are_separated implies X1 misses X2 & for Y being non empty
TopSpace, f1 being continuous Function of X1,Y, f2 being continuous Function of
  X2,Y holds f1 union f2 is continuous Function of X1 union X2,Y
  proof
    assume
A1: X1,X2 are_separated;
    hence X1 misses X2 by TSEP_1:63;
    X1,X2 are_weakly_separated by A1,TSEP_1:78;
    hence thesis by A1,Th133,TSEP_1:63;
  end;
  thus X1 misses X2 & (for Y being non empty TopSpace, f1 being continuous
  Function of X1,Y, f2 being continuous Function of X2,Y holds f1 union f2 is
  continuous Function of X1 union X2,Y) implies X1,X2 are_separated
  proof
    assume
A2: X1 misses X2;
    assume
A3: for Y being non empty TopSpace, f1 being continuous Function of X1
    ,Y, f2 being continuous Function of X2,Y holds f1 union f2 is continuous
    Function of X1 union X2,Y;
    now
      let Y be non empty TopSpace, g be Function of X1 union X2,Y;
      assume that
A4:   g|X1 is continuous Function of X1,Y and
A5:   g|X2 is continuous Function of X2,Y;
      reconsider f2 = g|X2 as continuous Function of X2,Y by A5;
      reconsider f1 = g|X1 as continuous Function of X1,Y by A4;
      g = f1 union f2 by Th126;
      hence g is continuous Function of X1 union X2,Y by A3;
    end;
    hence thesis by A2,Th123;
  end;
end;
