theorem
  F is associative implies
    F[:](id D,d)*(F.:(T1,T2)) = F.:(T1,F[:](id D,d)*T2 )
proof
  assume
A1: F is associative;
  per cases;
  suppose
A2: i = 0;
    then F.:(T1,T2) = <*>D by Lm1;
    then
A3: F[:](id D,d)*(F.:(T1,T2)) = <*>D;
    T1 = <*>D by A2;
    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,Th25;
  end;
end;
