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;

theorem Th117:
  X1,X2 are_weakly_separated implies for f being Function of X,Y
  holds f|(X1 union X2) is continuous Function of X1 union X2,Y iff f|X1 is
  continuous Function of X1,Y & f|X2 is continuous Function of X2,Y
proof
  assume
A1: X1,X2 are_weakly_separated;
  let f be Function of X,Y;
A2: X2 is SubSpace of X1 union X2 by TSEP_1:22;
  then
A3: f|(X1 union X2)|X2 = f|X2 by Th71;
A4: X1 is SubSpace of X1 union X2 by TSEP_1:22;
  then
A5: f|(X1 union X2)|X1 = f|X1 by Th71;
  hence
  f|(X1 union X2) is continuous Function of X1 union X2,Y implies f|X1 is
  continuous Function of X1,Y & f|X2 is continuous Function of X2,Y by A4,A2,A3
,Th82;
  thus thesis by A1,A5,A3,Th114;
end;
