reserve x,y for object,X,Y for set,
  D for non empty set,
  i,j,k,l,m,n,m9,n9 for Nat,
  i0,j0,n0,m0 for non zero Nat,
  K for Field,
  a,b for Element of K,
  p for FinSequence of K,
  M for Matrix of n,K;
reserve A for (Matrix of D),
  A9 for Matrix of n9,m9,D,
  M9 for Matrix of n9, m9,K,
  nt,nt1,nt2 for Element of n-tuples_on NAT,
  mt,mt1 for Element of m -tuples_on NAT,
  M for Matrix of K;
reserve P,P1,P2,Q,Q1,Q2 for without_zero finite Subset of NAT;
reserve v,v1,v2,u,w for Vector of n-VectSp_over K,
  t,t1,t2 for Element of n -tuples_on the carrier of K,

  L for Linear_Combination of n-VectSp_over K,
  M,M1 for Matrix of m,n,K;

theorem
  dim (n-VectSp_over K) = n
proof
  set ONE=1.(K,n);
  len ONE=n by MATRIX_0:24;
  then
A1: dom ONE=Seg n by FINSEQ_1:def 3;
  then
A2: ONE.:Seg n=lines ONE by RELAT_1:113;
  the_rank_of ONE=n by Lm8;
  then ONE is without_repeated_line by Th105;
  then Seg n,ONE.:Seg n are_equipotent by A1,CARD_1:33;
  then card Seg n=card lines ONE by A2,CARD_1:5;
  then
A3: card lines ONE=n by FINSEQ_1:57;
  lines ONE is Basis of n-VectSp_over K by Lm8;
  hence thesis by A3,VECTSP_9:def 1;
end;
