reserve x, y for object, X for set,
  i, j, k, l, n, m for Nat,
  D for non empty set,
  K for commutative Ring,
  a,b for Element of K,
  perm, p, q for Element of Permutations(n),
  Perm,P for Permutation of Seg n,
  F for Function of Seg n,Seg n,
  perm2, p2, q2, pq2 for Element of Permutations(n+2),
  Perm2 for Permutation of Seg (n+2);
reserve s for Element of 2Set Seg (n+2);

theorem Th22:
  K is non degenerated well-unital domRing-like
  implies
  (K is Fanoian iff 1_K <> -1_K)
proof
  assume A0: K is non degenerated well-unital domRing-like;
  thus K is Fanoian implies 1_K <> -1_K
  proof
    assume
A1: K is Fanoian;
    assume 1_K=-1_K;
    then 1_K+1_K=0.K by RLVECT_1:def 10;
    hence thesis by A0,A1;
  end;
  assume
A2: 1_K <> -1_K;
  assume not K is Fanoian;
  then consider a being Element of K such that
A3: a+a=0.K and
A4: a<>0.K;
  a=a*1_K;
  then 0.K=a*(1_K+1_K) by A3,VECTSP_1:def 7;
  then 0.K=1_K+1_K by A0,A4,VECTSP_2:def 1;
  hence thesis by A2,VECTSP_1:16;
end;
