reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve c for Complex;

theorem Th13:
  113 is prime
  proof
    for n being Element of NAT holds 1 < n & n*n <= 113 & n is prime implies
    not n divides 113
    proof
      let n be Element of NAT;
      113 = 2*56 + 1;
      then
A1:   not 2 divides 113;
      113 = 3*37 + 2;
      then
A2:   not 3 divides 113 by NAT_4:9;
      113 = 5*22 + 3;
      then
A3:   not 5 divides 113 by NAT_4:9;
      113 = 7*16 + 1;
      then
A4:   not 7 divides 113 by NAT_4:9;
      113 = 11*10 + 3;
      then
A5:   not 11 divides 113 by NAT_4:9;
      113 = 13*8 + 9;
      then
A6:   not 13 divides 113 by NAT_4:9;
      113 = 17*6 + 11;
      then
A7:   not 17 divides 113 by NAT_4:9;
      113 = 19*5 + 18;
      then
A8:   not 19 divides 113 by NAT_4:9;
      113 = 23*4 + 21;
      then
      not 23 divides 113 by NAT_4:9;
      hence thesis by A1,A2,A3,A4,A5,A6,A7,A8,NAT_4:62;
    end;
    hence thesis by NAT_4:14;
  end;
