
theorem
  7537 is prime
proof
  now
    7537 = 2*3768 + 1; hence not 2 divides 7537 by NAT_4:9;
    7537 = 3*2512 + 1; hence not 3 divides 7537 by NAT_4:9;
    7537 = 5*1507 + 2; hence not 5 divides 7537 by NAT_4:9;
    7537 = 7*1076 + 5; hence not 7 divides 7537 by NAT_4:9;
    7537 = 11*685 + 2; hence not 11 divides 7537 by NAT_4:9;
    7537 = 13*579 + 10; hence not 13 divides 7537 by NAT_4:9;
    7537 = 17*443 + 6; hence not 17 divides 7537 by NAT_4:9;
    7537 = 19*396 + 13; hence not 19 divides 7537 by NAT_4:9;
    7537 = 23*327 + 16; hence not 23 divides 7537 by NAT_4:9;
    7537 = 29*259 + 26; hence not 29 divides 7537 by NAT_4:9;
    7537 = 31*243 + 4; hence not 31 divides 7537 by NAT_4:9;
    7537 = 37*203 + 26; hence not 37 divides 7537 by NAT_4:9;
    7537 = 41*183 + 34; hence not 41 divides 7537 by NAT_4:9;
    7537 = 43*175 + 12; hence not 43 divides 7537 by NAT_4:9;
    7537 = 47*160 + 17; hence not 47 divides 7537 by NAT_4:9;
    7537 = 53*142 + 11; hence not 53 divides 7537 by NAT_4:9;
    7537 = 59*127 + 44; hence not 59 divides 7537 by NAT_4:9;
    7537 = 61*123 + 34; hence not 61 divides 7537 by NAT_4:9;
    7537 = 67*112 + 33; hence not 67 divides 7537 by NAT_4:9;
    7537 = 71*106 + 11; hence not 71 divides 7537 by NAT_4:9;
    7537 = 73*103 + 18; hence not 73 divides 7537 by NAT_4:9;
    7537 = 79*95 + 32; hence not 79 divides 7537 by NAT_4:9;
    7537 = 83*90 + 67; hence not 83 divides 7537 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 7537 & n is prime
  holds not n divides 7537 by XPRIMET1:46;
  hence thesis by NAT_4:14;
end;
