theorem
  F is associative & F is having_a_unity & F is having_an_inverseOp &
  G is_distributive_wrt F implies
    G[:](id D,d).the_unity_wrt F = the_unity_wrt F
proof
  assume that
A1: F is associative and
A2: F is having_a_unity and
A3: F is having_an_inverseOp and
A4: G is_distributive_wrt F;
  set e = the_unity_wrt F, i = the_inverseOp_wrt F;
  G.(e,d) = G.(F.(e,e),d) by A2,SETWISEO:15
    .= F.(G.(e,d),G.(e,d)) by A4,BINOP_1:11;
  then e = F.(F.(G.(e,d),G.(e,d)),i.(G.(e,d))) by A1,A2,A3,Th59;
  then e = F.(G.(e,d),F.(G.(e,d),i.(G.(e,d)))) by A1;
  then e = F.(G.(e,d),e) by A1,A2,A3,Th59;
  then e = G.(e,d) by A2,SETWISEO:15;
  then G.((id D).e,d) = e;
  hence thesis by FUNCOP_1:48;
end;
