reserve i,j,n for Nat,
  K for Field,
  a for Element of K,
  M,M1,M2,M3,M4 for Matrix of n,K;
reserve A for Matrix of K;

theorem
  for K being Fanoian Field,n being Nat,M1 being Matrix of n,K st 
  M1 is symmetric antisymmetric holds M1=0.(K,n,n)
proof
  let K be Fanoian Field;
  let n;
  let M1 be Matrix of n,K;
  assume M1 is symmetric antisymmetric;
  then
A1: M1@=M1 & M1@=-M1;
  for i,j st [i,j] in Indices M1 holds M1*(i,j)=(0.(K,n,n))*(i,j)
  proof
    let i,j;
    assume
A2: [i,j] in Indices M1;
    then M1*(i,j)=-M1*(i,j) by A1,MATRIX_3:def 2;
    then M1*(i,j)+M1*(i,j)=0.K by RLVECT_1:5;
    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;
    then
A3: M1*(i,j)=0.K by VECTSP_1:12;
    [i,j] in Indices (0.(K,n,n)) by A2,MATRIX_0:26;
    hence thesis by A3,MATRIX_3:1;
  end;
  hence thesis by MATRIX_0:27;
end;
