theorem
  F is associative & F is having_a_unity & F is having_an_inverseOp &
  G is_distributive_wrt F implies
    G[;](d,id 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.(d,e) = G.(d,F.(e,e)) by A2,SETWISEO:15
    .= F.(G.(d,e),G.(d,e)) by A4,BINOP_1:11;
  then e = F.(F.(G.(d,e),G.(d,e)),i.(G.(d,e))) by A1,A2,A3,Th59;
  then e = F.(G.(d,e),F.(G.(d,e),i.(G.(d,e)))) by A1;
  then e = F.(G.(d,e),e) by A1,A2,A3,Th59;
  then e = G.(d,e) by A2,SETWISEO:15;
  then G.(d,(id D).e) = e;
  hence thesis by FUNCOP_1:53;
end;
