reserve x,y,z,X,Y for set;
reserve X,Y for non empty set,
  f for Function of X,Y;
reserve X, Y for non empty set,
  F for (BinOp of Y),
  B for (Element of Fin X),
  f for Function of X,Y;
reserve A for set,
  x,y,z for Element of Fin A;
reserve X,Y for non empty set,
  A for set,
  f for (Function of X, Fin A),
  i,j,k for (Element of X);

theorem
  for B being Element of Fin X for f being Function of X,Y holds for g
  being Function of Y,Fin A holds FinUnion(f.:B,g) = FinUnion(B,g*f)
proof
  let B be Element of Fin X;
  let f be Function of X,Y;
  let g be Function of Y,Fin A;
  thus FinUnion(f.:B,g) = union (g.:(f.:B)) by Th46
    .= union ((g*f).:B) by RELAT_1:126
    .= FinUnion(B,g*f) by Th46;
end;
