theorem
  F is associative implies F.:(F[:](T1,d),T2) = F.:(T1,F[;](d,T2))
proof
  assume
A1: F is associative;
  per cases;
  suppose
A2: i = 0;
    then F[:](T1,d) = <*>D by Lm3;
    then
A3: F.:(F[:](T1,d),T2) = <*>D by FINSEQ_2:73;
    F[;](d,T2) = <*>D by A2,Lm2;
    hence thesis by A3,FINSEQ_2:73;
  end;
  suppose
    i <> 0;
    then reconsider C = Seg i as non empty set;
    T1 is Function of C,D & T2 is Function of C,D by Lm4;
    hence thesis by A1,FUNCOP_1:60;
  end;
end;
