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 st X = X1 union X2 for g
  being Function of X,Y holds g = (g|X1) union (g|X2)
proof
  let X1, X2 be non empty SubSpace of X such that
A1: X = X1 union X2;
  let g be Function of X,Y;
  reconsider h = g as Function of X1 union X2,Y by A1;
  X2 is SubSpace of X1 union X2 by TSEP_1:22;
  then
A2: h|X2 = g|X2 by Def5;
  X1 is SubSpace of X1 union X2 by TSEP_1:22;
  then h|X1 = g|X1 by Def5;
  hence thesis by A2,Th126;
end;
