 reserve n,s for Nat;

theorem Th46:
  n <> 2 implies Triangle n is non prime
  proof
    assume
A1: n <> 2;
    assume
A2: Triangle n is prime; then
A3: n <> 1 by Th11,INT_2:def 4;
    n <> 0 by A2,INT_2:def 4; then
A4: n is non trivial by NAT_2:def 1,A3;
    per cases;
    suppose n is odd; then
      n = 1 or n = Triangle n by INT_2:def 4,A2,Th41;
      hence thesis by Th45,A4,Th11;
    end;
    suppose n is even; then
      consider k being Nat such that
A5:   n = 2 * k by ABIAN:def 2;
A6:   k <> 0 by A2,INT_2:def 4,A5;
A7:   Triangle n = n * (n + 1) / 2 by Th19
                .= k * (n + 1) by A5; then
      k divides Triangle n by NAT_D:def 3; then
      per cases by INT_2:def 4,A2;
      suppose k = 1;
        hence thesis by A1,A5;
      end;
      suppose
A8:     k = Triangle n;
        1 = k / k by A6,XCMPLX_1:60
         .= n + 1 by A6,XCMPLX_1:89,A7,A8; then
        n = 0;
        hence thesis by INT_2:def 4,A2;
      end;
    end;
  end;
