reserve

  k,n,m,i,j for Element of NAT,
  K for Field;
reserve L for non empty addLoopStr;
reserve G for non empty multLoopStr;

theorem Th45:
  for n being Element of NAT, A being Matrix of n,K for i,j being
Nat st 1<=i & i<=n & 1<=j & j<=n holds (i<>j implies SwapDiagonal(K,n,1)*(i,j)=
  0.K)
proof
  let n be Element of NAT, A be Matrix of n,K;
  set A= SwapDiagonal(K,n,1);
  let i,j be Nat;
  assume 1<=i & i<=n & 1<=j & j<=n;
  then
A1: [i,j] in Indices A by MATRIX_0:31;
  A=1.(K,n) by FINSEQ_7:19;
  hence thesis by A1,MATRIX_1:def 3;
end;
