
theorem Th31:
  37 is prime
proof
  now
    let n be Element of NAT;
    37 = 2*18 + 1;
    then
A1: not 2 divides 37 by Th9;
    37 = 3*12 + 1;
    then
A2: not 3 divides 37 by Th9;
    37 = 13*2 + 11;
    then
A3: not 13 divides 37 by Th9;
    37 = 11*3 + 4;
    then
A4: not 11 divides 37 by Th9;
    37 = 19*1 + 18;
    then
A5: not 19 divides 37 by Th9;
    37 = 17*2 + 3;
    then
A6: not 17 divides 37 by Th9;
    37 = 23*1 + 14;
    then
A7: not 23 divides 37 by Th9;
    37 = 7*5 + 2;
    then
A8: not 7 divides 37 by Th9;
    37 = 5*7 + 2;
    then
A9: not 5 divides 37 by Th9;
    assume 1<n & n*n<=37 & n is prime;
    hence not n divides 37 by A1,A2,A9,A8,A4,A3,A6,A5,A7,Lm6;
  end;
  hence thesis by Th14;
end;
