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,i being Nat,M1 being Matrix of n,K st 
  M1 is antisymmetric & i in Seg n holds M1*(i,i) = 0.K
proof
  let K be Fanoian Field;
  let n,i;
  let M1 be Matrix of n,K;
  assume that
A1: M1 is antisymmetric and
A2: i in Seg n;
A3: M1@=-M1 by A1;
  Indices M1=[:Seg n, Seg n:] by MATRIX_0:24;
  then
A4: [i,i] in Indices M1 by A2,ZFMISC_1:87;
  then M1@*(i,i)=M1*(i,i) by MATRIX_0:def 6;
  then M1*(i,i)=-M1*(i,i) by A4,A3,MATRIX_3:def 2;
  then M1*(i,i)+M1*(i,i)=0.K by RLVECT_1:5;
  then (1_K)*(M1*(i,i))+(1_K)*(M1*(i,i))=0.K;
  then 1_K+1_K<>0.K & (1_K+1_K)*(M1*(i,i))=0.K by VECTSP_1:def 7,def 19;
  hence thesis by VECTSP_1:12;
end;
