theorem Th56:
   for s be Element of S holds
   f is RingHomomorphism & f.:S c= Unit_Set(B) implies f.s is Unit of B
   proof
     let s be Element of S;
     assume
A1:  f is RingHomomorphism & f.:S c= Unit_Set(B);
A2:  dom f = the carrier of A by FUNCT_2:def 1;
     reconsider t = f.s as object;
     t in f.:S by A2,FUNCT_1:def 6; then
     1.B = f.s*((f.s)["]) by A1, Def2; then
     f.s divides 1.B; then
     f.s is unital;
     hence thesis;
   end;
