reserve x,y for object, X for set;

theorem Th21:
  for p be Prime holds 1_Z/Z*(p) =1 & 1_Z/Z*(p) = 1.(INT.Ring(p))
proof
  let p be Prime;
A1: not 1 in {0} by TARSKI:def 1;
A2: 1 < p by INT_2:def 4;
  then 1 in Segm(p) by NAT_1:44;
  then 1 in Segm(p)\{0} by A1,XBOOLE_0:def 5;
  then 1 in Segm0(p) by A2,Def2;
  then reconsider e=1.(INT.Ring(p)) as Element of Z/Z*(p) by A2,INT_3:14;
  now
    let h being Element of Z/Z*(p);
    h in Segm0(p);
    then h in Segm(p)\{0} by A2,Def2;
    then reconsider hp=h as Element of INT.Ring(p) by XBOOLE_0:def 5;
    thus h * e = hp*1_(INT.Ring(p)) by Lm12
      .= h;
    thus e * h = 1_(INT.Ring(p))*hp by Lm12
      .= h;
  end;
  then e=1_(Z/Z*(p)) by GROUP_1:def 4;
  hence thesis by A2,INT_3:14;
end;
