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 Th49:
  for A,X being non empty set, Y being set for G being BinOp of A
st G is commutative & G is associative & G is idempotent for B being Element of
Fin X st B <> {} for f being (Function of X,Fin Y), g being Function of Fin Y,A
  st for x,y being Element of Fin Y holds g.(x \/ y) = G.(g.x,g.y) holds g.(
  FinUnion(B,f)) = G$$(B,g*f)
proof
  let A,X be non empty set, Y be set, G be BinOp of A such that
A1: G is commutative & G is associative & G is idempotent;
  let B be Element of Fin X such that
A2: B <> {};
  let f be (Function of X,Fin Y), g be Function of Fin Y,A such that
A3: for x,y being Element of Fin Y holds g.(x \/ y) = G.(g.x,g.y);
A4: now
    let x,y be Element of Fin Y;
    thus g.(FinUnion Y.(x,y)) = g.(x \/ y) by Def4
      .= G.(g.x,g.y) by A3;
  end;
A5: FinUnion Y is idempotent by Th34;
  FinUnion Y is commutative & FinUnion Y is associative by Th35,Th36;
  hence thesis by A1,A5,A2,A4,Th27;
end;
