theorem
  F is associative implies F[:](F[;](d1,T),d2) = F[;](d1,F[:](T,d2))
proof
  assume
A1: F is associative;
  per cases;
  suppose
A2: i = 0;
    then F[;](d1,T) = <*>D by Lm2;
    then
A3: F[:](F[;](d1,T),d2) = <*>D by FINSEQ_2:85;
    F[:](T,d2) = <*>D by A2,Lm3;
    hence thesis by A3,FINSEQ_2:79;
  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_1:59;
  end;
end;
