theorem
  F is having_a_unity & F is associative & F is having_an_inverseOp &
  u = the_inverseOp_wrt F & G is_distributive_wrt F & G is having_a_unity
  implies G [;](u.(the_unity_wrt G),id D) = u
proof
  assume that
A1: F is having_a_unity & F is associative & F is having_an_inverseOp &
  u = the_inverseOp_wrt F & G is_distributive_wrt F and
A2: G is having_a_unity;
  set o = the_unity_wrt G;
  for d holds (G[;](u.o,id D)).d = u.d
  proof
    let d;
    thus (G[;](u.o,id D)).d = G.(u.o,(id D).d) by FUNCOP_1:53
      .= G.(u.o,d)
      .= u.(G.(o,d)) by A1,Th67
      .= u.d by A2,SETWISEO:15;
  end;
  hence thesis by FUNCT_2:63;
end;
