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 Th59:
  f is RingHomomorphism & f.:S c= Unit_Set(B) implies
    Univ_Map(S,f) is unity-preserving
   proof
     set F = Univ_Map(S,f);
     assume
A1:  f is RingHomomorphism & f.:S c= Unit_Set(B);
     Univ_Map(S,f).1.(S~A)= 1.B
     proof
       consider a1,s1 being Element of A such that
A3:    s1 in S and
A4:    1.(S~A) = Class(EqRel(S),[a1,s1]) and
A5:    F.1.(S~A) = (f.a1)*((f.s1)["]) by A1,Def8;
       reconsider as1 = [a1,s1] as Element of Frac(S) by A3,Def3;
       Class(EqRel(S),1.(A,S)) = Class(EqRel(S),as1) by A4,Def6; then
A6:    as1, 1.(A,S) Fr_Eq S by Th26;
       consider s0 being Element of A such that
A7:    s0 in S and
A8:    (as1`1 * 1.(A,S)`2 - 1.(A,S)`1 * as1`2) * s0 = 0.A by A6;
A9:    0.B = f.(0.A) by A1,QUOFIELD:50
       .= f.(a1 - s1)*f.s0 by A1,A8,GROUP_6:def 6;
       f.s0 is Unit of B by A1,A7,Th56; then
A10:   f.s0 in Unit_Set(B);
A11:   0.B = (f.(a1 - s1)*f.s0)*((f.s0)["]) by A9
       .= f.(a1 - s1)*(f.s0*((f.s0)["])) by GROUP_1:def 3
       .= f.(a1 - s1)*(1.B) by A10,Def2
       .= f.a1 - f.s1 by A1,RING_2:8;
       f.s1 is Unit of B by A1,A3,Th56; then
A12:   f.s1 in Unit_Set(B);
       Univ_Map(S,f).1.(S~A)
       = (f.s1)*((f.s1)["]) by A5,A11,VECTSP_1:27 .= 1.B by A12,Def2;
       hence thesis;
     end;
     hence thesis;
   end;
