theorem
  F is associative implies F[:](T,F.(d1,d2)) = F[:](F[:](T,d1),d2)
proof
  assume
A1: F is associative;
  per cases;
  suppose
    i = 0;
    then T = <*>D & F[:](T,d1) = <*>D by Lm3;
    hence thesis;
  end;
  suppose
    i <> 0;
    then reconsider C = Seg i as non empty set;
    T is Function of C,D by Lm4;
    hence thesis by A1,FUNCOP_1:63;
  end;
end;
