
theorem Th37:
  631 is prime
proof
  now
    let n be Element of NAT;
    631 = 2*315 + 1;
    then
A1: not 2 divides 631 by Th9;
    631 = 3*210 + 1;
    then
A2: not 3 divides 631 by Th9;
    631 = 13*48 + 7;
    then
A3: not 13 divides 631 by Th9;
    631 = 11*57 + 4;
    then
A4: not 11 divides 631 by Th9;
    631 = 19*33 + 4;
    then
A5: not 19 divides 631 by Th9;
    631 = 17*37 + 2;
    then
A6: not 17 divides 631 by Th9;
    631 = 23*27 + 10;
    then
A7: not 23 divides 631 by Th9;
    631 = 7*90 + 1;
    then
A8: not 7 divides 631 by Th9;
    631 = 5*126 + 1;
    then
A9: not 5 divides 631 by Th9;
    assume 1<n & n*n<=631 & n is prime;
    hence not n divides 631 by A1,A2,A9,A8,A4,A3,A6,A5,A7,Lm6;
  end;
  hence thesis by Th14;
end;
