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;
