reserve n,k,b for Nat, i for Integer;

theorem Th34:
  89 is prime
  proof
    now
      let n be Element of NAT;
      89 = 2*44 + 1;
      then
      A1: not 2 divides 89 by NAT_4:9;
      89 = 3*29 + 2;
      then
      A2: not 3 divides 89 by NAT_4:9;
      89 = 5*17 + 4;
      then
      A3: not 5 divides 89 by NAT_4:9;
      89 = 7*12 + 5;
      then
      A4: not 7 divides 89 by NAT_4:9;
      89 = 11*8 + 1;
      then
      A5: not 11 divides 89 by NAT_4:9;
      89 = 13*6 + 11;
      then
      A6: not 13 divides 89 by NAT_4:9;
      89 = 17*5 + 4;
      then
      A7: not 17 divides 89 by NAT_4:9;
      89 = 19*4 + 13;
      then
      A8: not 19 divides 89 by NAT_4:9;
      89 = 23*3 + 20;
      then
      A9: not 23 divides 89 by NAT_4:9;
      assume 1<n & n*n<=89 & n is prime;
      hence not n divides 89 by A1,A2,A3,A4,A5,A6,A7,A8,A9,NAT_4:62;
    end;
    hence thesis by NAT_4:14;
  end;
