theorem Th25:
  F is associative implies F[:](id D,d)*(F.:(f,f9)) = F.:(f,F[:](id D,d)*f9)
proof
  assume
A1: F is associative;
  now
    let c;
    thus (F[:](id D,d)*(F.:(f,f9))).c = (F[:](id D,d)).((F.:(f,f9)).c) by
FUNCT_2:15
      .= (F[:](id D,d)).(F.(f.c,f9.c)) by FUNCOP_1:37
      .= F.((id D).(F.(f.c,f9.c)),d) by FUNCOP_1:48
      .= F.(F.(f.c,f9.c),d)
      .= F.(f.c,F.(f9.c,d)) by A1
      .= F.(f.c,(F[:](f9,d)).c) by FUNCOP_1:48
      .= F.(f.c,((F[:](id D,d))*f9).c) by FUNCOP_1:50
      .= (F.:(f,F[:](id D,d)*f9)).c by FUNCOP_1:37;
  end;
  hence thesis by FUNCT_2:63;
end;
