theorem Th66:
  F is associative & F is having_a_unity & F is having_an_inverseOp &
  G is_distributive_wrt F & e = the_unity_wrt F implies
  for d holds G.(e,d) = e & G.(d,e) = e
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 and
A5: e = the_unity_wrt F;
  let d;
  set ed = G.(e,d);
  F.(ed,ed) = G.(F.(e,e),d) by A4,BINOP_1:11
    .= ed by A2,A5,SETWISEO:15;
  hence ed = e by A1,A2,A3,A5,Th65;
  set de = G.(d,e);
  F.(de,de) = G.(d,F.(e,e)) by A4,BINOP_1:11
    .= de by A2,A5,SETWISEO:15;
  hence thesis by A1,A2,A3,A5,Th65;
end;
