 reserve o for object;
 reserve F for non almost_trivial Field;
 reserve x,a for Element of F;
reserve n for non zero Nat;

theorem Th19:
   for R being Ring, p being Polynomial of R for i being Integer holds p <> i
   proof
     let R be Ring, p be Polynomial of R; let i be Integer;
A1:  i in INT by INT_1:def 2;
     now assume
A2:   p = i;
      per cases by A1,NUMBERS:def 4,XBOOLE_0:def 3;
        suppose i in NAT; then
          reconsider n = i as Element of NAT;
          p = n by A2;
          hence contradiction by Th18;
        end;
        suppose i in [:{0},NAT:]; then
          consider x,y being object such that
A3:       x in {0} & y in NAT & i = [x,y] by ZFMISC_1:def 2;
          reconsider n = y as Element of NAT by A3;
A4:       p = [0,n] by A2,A3,TARSKI:def 1 .= {{0,n},{0}} by TARSKI:def 5;
A5:       dom p = NAT by FUNCT_2:def 1;
          per cases by A4,A5;
            suppose
A6:           [0,p.0] = {0,n};
              [0,p.0] = {{0,p.0},{0}} by TARSKI:def 5; then
A7:           0 in {{0,p.0},{0}} by A6,TARSKI:def 2;
              per cases by A7;
                suppose 0 = {0};
                  hence contradiction by CARD_1:49;
                end;
                suppose 0 = {0,p.0}; then
                  {} = {0,p.0};
                  hence contradiction;
                end;
              end;
              suppose [0,p.0] = {0};
                hence contradiction by CARD_1:49;
              end;
            end;
          end;
          hence thesis;
        end;
