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;

theorem Th12:
  for u be Unit of R, v be Element of R holds
  f is RingHomomorphism implies f.u is Unit of R1 & (f.u)["] = f.(u["])
  proof
    let u be Unit of R, v be Element of R;
    assume
A1: f is RingHomomorphism; then
A3: f is multiplicative;
A2: u in Unit_Set(R);
    f is unity-preserving by A1; then
A5: 1.R1 = f.(u*(u["])) by A2,Def2
      .= f.u*f.(u["]) by A3; then
    f.u divides 1.R1; then
A7: f.u is unital; then
A8: f.u in Unit_Set(R1);
    (f.u)["] = (f.u)["]*(f.u* f.(u["])) by A5
      .= ((f.u)["]*f.u)* f.(u["]) by GROUP_1:def 3
      .= 1.R1*f.(u["]) by Def2,A8
      .= f.(u["]);
    hence thesis by A7;
  end;
