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 Y,Z being set for f being Function of X, Fin Y for g being
  Function of Fin Y, Fin Z st g.{}.Y = {}.Z & for x,y being Element of Fin Y
holds g.(x \/ y) = g.x \/ g.y for B being Element of Fin X holds g.(FinUnion(B,
  f)) = FinUnion(B,g*f)
proof
  let Y,Z be set;
  let f be Function of X, Fin Y;
  let g be Function of Fin Y, Fin Z;
  assume that
A1: g.{}.Y = {}.Z and
A2: for x,y being Element of Fin Y holds g.(x \/ y) = g.x \/ g.y;
A3: g.{}.Y = the_unity_wrt FinUnion Z by A1,Th40;
A4: now
    let x,y be Element of Fin Y;
    thus g.(x \/ y) = g.x \/ g.y by A2
      .= FinUnion Z.(g.x,g.y) by Def4;
  end;
  let B be Element of Fin X;
A5: FinUnion Z is idempotent by Th34;
  FinUnion Z is associative & FinUnion Z is commutative by Th35,Th36;
  hence thesis by A5,A3,A4,Th38,Th50;
end;
