theorem
  F is associative implies F[;](F.(d1,d2),T) = F[;](d1,F[;](d2,T))
proof
  assume
A1: F is associative;
  per cases;
  suppose
    i = 0;
    then T = <*>D & F[;](d2,T) = <*>D by Lm2;
    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_1:62;
  end;
end;
