
theorem ass:
for X being non empty set
for L being non empty add-associative addLoopStr
for f,g,h being Function of X,L holds (f '+' g) '+' h = f '+' (g '+' h)
proof
let X be non empty set, L be non empty add-associative addLoopStr,
    f,g,h be Function of X,L;
now let o be object;
  assume o in X; then
  reconsider x = o as Element of X;
  thus ((f '+' g) '+' h).o
     = (f'+'g).x + h.x by defp
    .= (f.x + g.x) + h.x by defp
    .= f.x + (g.x + h.x) by RLVECT_1:def 3
    .= f.x + (g'+'h).x by defp
    .= (f '+' (g '+' h)).o by defp;
  end;
hence thesis by FUNCT_2:12;
end;
