reserve i, j, m, n, k for Nat,
  x, y for set,
  K for Field,
  a,a1 for Element of K;
reserve V1,V2,V3 for finite-dimensional VectSp of K,
  f for Function of V1,V2,

  b1,b19 for OrdBasis of V1,
  B1 for FinSequence of V1,
  b2 for OrdBasis of V2,
  B2 for FinSequence of V2,

  B3 for FinSequence of V3,
  v1,w1 for Element of V1,
  R,R1,R2 for FinSequence of V1,
  p,p1,p2 for FinSequence of K;

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;
