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 Th102:
  the carrier of n -VectSp_over K = n-tuples_on the carrier of K
& 0.(n -VectSp_over K) = n |-> 0.K & ( t1 = v1 & t2 = v2 implies t1 + t2 = v1 +
  v2 ) & ( t = v implies a * t = a * v )
proof
A1: the addLoopStr of n -VectSp_over K=n-Group_over K by PRVECT_1:def 5;
A2: n-Group_over K=addLoopStr(# n-tuples_on the carrier of K, product(the
    addF of K,n), (n |-> 0.K) qua Element of n-tuples_on the carrier of K#)
by PRVECT_1:def 3;
  hence the carrier of n -VectSp_over K = n-tuples_on the carrier of K & 0.(n
  -VectSp_over K) =(n |-> 0.K) by A1;
  thus t1 = v1 & t2 = v2 implies t1 + t2 = v1 + v2 by A2,A1,PRVECT_1:def 1;
  assume
A3: t=v;
  rng t c= the carrier of K by RELAT_1:def 19;
  then
A4: (id (the carrier of K))*t=t by RELAT_1:53;
  thus a*v = (n-Mult_over K).(a,v) by PRVECT_1:def 5
    .= (the multF of K)[;](a,t) by A3,PRVECT_1:def 4
    .= a*t by A4,FUNCOP_1:34;
end;
