reserve x,y for object, X for set;

theorem
  for p be Prime, x be Element of Z/Z*(p), x1 be Element of INT.Ring(p)
  st x=x1 holds x" = x1"
proof
  let p be Prime, h be Element of Z/Z*(p), hp be Element of INT.Ring(p);
  assume
A1: h=hp;
A2: 0 in Segm(p) by NAT_1:44;
  set hpd=hp";
A3: 1 < p by INT_2:def 4;
  h in Segm0(p);
  then h in Segm(p)\{0} by A3,Def2;
  then not h in {0} by XBOOLE_0:def 5;
  then hp <> 0 by A1,TARSKI:def 1;
  then
A4: hp <> 0.(INT.Ring(p)) by A2,SUBSET_1:def 8;
  then hp*hpd =1.(INT.Ring(p)) by VECTSP_1:def 10;
  then hpd <> 0.(INT.Ring(p));
  then hpd <> 0;
  then not hpd in {0} by TARSKI:def 1;
  then hpd in Segm(p)\{0} by XBOOLE_0:def 5;
  then reconsider g=hpd as Element of Z/Z*(p) by A3,Def2;
  h*g=hp*hpd by A1,Lm12
    .=1.(INT.Ring(p)) by A4,VECTSP_1:def 10
    .=1_(Z/Z*(p)) by Th21;
  hence thesis by GROUP_1:def 5;
end;
