theorem Th45:
  MX2FinS 1.(K,n) is OrdBasis of n-VectSp_over K
proof
  set ONE=1.(K,n);
A1: the_rank_of ONE=n by Lm6;
  then
A2: ONE is one-to-one by MATRIX13:105;
  for i,j st [i,j] in Indices ONE & ONE*(i,j) <> 0.K holds i=j by
MATRIX_1:def 3;
  then ONE is diagonal by MATRIX_1:def 6;
  then lines ONE is Basis of n-VectSp_over K by A1,MATRIX13:111;
  hence thesis by A2,MATRLIN:def 2;
end;
