reserve R,R1 for commutative Ring;
reserve A,B for non degenerated commutative Ring;
reserve o,o1,o2 for object;
reserve r,r1,r2 for Element of R;
reserve a,a1,a2,b,b1 for Element of A;
reserve f for Function of R, R1;
reserve p for Element of Spectrum A;
reserve S for non empty multiplicatively-closed Subset of R;
reserve u,v,w,x,y,z for Element of Frac(S);
reserve a, b, c for Element of Frac(S);
reserve x, y, z for Element of S~R;
reserve S for without_zero non empty multiplicatively-closed Subset of A;
reserve p for Element of Spectrum A;
reserve a,m,n for Element of A~p;
reserve f for Function of A,B;

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;
