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 Th44:
  for n being Element of NAT, A being Matrix of n,K for i being
  Nat st 1<=i & i<=n holds SwapDiagonal(K,n,1)*(i,i)=1.K
proof
  let n be Element of NAT, A be Matrix of n,K;
  set A= SwapDiagonal(K,n,1);
  let i be Nat;
  assume 1<=i & i<=n;
  then A=1.(K,n) & [i,i] in Indices A by FINSEQ_7:19,MATRIX_0:31;
  hence thesis by MATRIX_1:def 3;
end;
