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

theorem Th24:
  31 is prime
  proof
    now
      let n be Element of NAT;
      31 = 2*15 + 1;
      then
      A1: not 2 divides 31 by NAT_4:9;
      31 = 3*10 + 1;
      then
      A2: not 3 divides 31 by NAT_4:9;
      31 = 5*6 + 1;
      then
      A3: not 5 divides 31 by NAT_4:9;
      31 = 7*4 + 3;
      then
      A4: not 7 divides 31 by NAT_4:9;
      31 = 11*2 + 9;
      then
      A5: not 11 divides 31 by NAT_4:9;
      31 = 13*2 + 5;
      then
      A6: not 13 divides 31 by NAT_4:9;
      31 = 17*1 + 14;
      then
      A7: not 17 divides 31 by NAT_4:9;
      31 = 19*1 + 12;
      then
      A8: not 19 divides 31 by NAT_4:9;
      31 = 23*1 + 8;
      then
      A9: not 23 divides 31 by NAT_4:9;
      assume 1<n & n*n<=31 & n is prime;
      hence not n divides 31 by A1,A2,A9,A8,A4,A3,A6,A5,A7,NAT_4:62;
    end;
    hence thesis by NAT_4:14;
  end;
