
theorem Coprime21:
  for a being Nat holds
    a,2 * a + 1 are_coprime
  proof
    let a be Nat;
    assume not a,2 * a + 1 are_coprime; then
    consider n being Nat such that
A1: n divides a & n divides 2 * a + 1 & n <> 1 by PYTHTRIP:def 1;
A2: 1 divides n by NAT_D:6;
    n divides (2 * a) by A1,NAT_D:9; then
    n divides 1 by A1,NAT_D:10;
    hence thesis by A1,A2,NAT_D:5;
  end;
