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;
