
theorem
  1543 is prime
proof
  now
    1543 = 2*771 + 1; hence not 2 divides 1543 by NAT_4:9;
    1543 = 3*514 + 1; hence not 3 divides 1543 by NAT_4:9;
    1543 = 5*308 + 3; hence not 5 divides 1543 by NAT_4:9;
    1543 = 7*220 + 3; hence not 7 divides 1543 by NAT_4:9;
    1543 = 11*140 + 3; hence not 11 divides 1543 by NAT_4:9;
    1543 = 13*118 + 9; hence not 13 divides 1543 by NAT_4:9;
    1543 = 17*90 + 13; hence not 17 divides 1543 by NAT_4:9;
    1543 = 19*81 + 4; hence not 19 divides 1543 by NAT_4:9;
    1543 = 23*67 + 2; hence not 23 divides 1543 by NAT_4:9;
    1543 = 29*53 + 6; hence not 29 divides 1543 by NAT_4:9;
    1543 = 31*49 + 24; hence not 31 divides 1543 by NAT_4:9;
    1543 = 37*41 + 26; hence not 37 divides 1543 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 1543 & n is prime
  holds not n divides 1543 by XPRIMET1:24;
  hence thesis by NAT_4:14;
end;
