reserve i,j,k,n,l for Nat,
  K for Field,
  a,b,c for Element of K,
  p,q for FinSequence of K,
  M1,M2,M3 for Matrix of n,K;

theorem
  for K being Fanoian Field,n,i,j,k,l being Nat,M1 being Matrix of n,K st
  [i,j] in Indices M1 & i+j=n+1 & k=n+1-j & l=n+1-i &
  M1 is Anti-subsymmetric holds M1*(i,j)=0.K
  proof
     let K be Fanoian Field;
     let n,i,j,k,l be Nat;
     let M1 be Matrix of n,K;
     assume that
A1:[i,j] in Indices M1 and
A2:i+j=n+1 & k=n+1-j & l=n+1-i and
A3:M1 is Anti-subsymmetric;
      M1*(i,j) = -(M1*(i,j)) by A1,A3,A2;
     then M1*(i,j)+M1*(i,j)=0.K by RLVECT_1:5;
     then (1_K)*(M1*(i,j))+M1*(i,j)=0.K;
     then (1_K)*(M1*(i,j))+(1_K)*(M1*(i,j))=0.K;
     then 1_K+1_K<>0.K & (1_K+1_K)*(M1*(i,j))=0.K by VECTSP_1:def 7,def 19;
  hence thesis by VECTSP_1:12;
end;
