theorem
  F is associative implies F.:(F.:(T1,T2),T3) = F.:(T1,F.:(T2,T3))
proof
  assume
A1: F is associative;
  per cases;
  suppose
A2: i = 0;
    then F.:(T1,T2) = <*>D by Lm1;
    then
A3: F.:(F.:(T1,T2),T3) = <*>D by FINSEQ_2:73;
    F.:(T2,T3) = <*>D by A2,Lm1;
    hence thesis by A3,FINSEQ_2:73;
  end;
  suppose
    i <> 0;
    then reconsider C = Seg i as non empty set;
A4: T3 is Function of C,D by Lm4;
    T1 is Function of C,D & T2 is Function of C,D by Lm4;
    hence thesis by A1,A4,FUNCOP_1:61;
  end;
end;
