
theorem lemspan:
for R being non degenerated commutative Ring, n being Nat
holds Lin Base(R,n) = R^*n
proof
let R be non degenerated commutative Ring, n be Nat;
set V = Lin Base(R,n), B = Base(R,n);
A: the carrier of V = the set of all Sum(l) where
                      l is Linear_Combination of B by VECTSP_7:def 2;
B: the carrier of V c= the carrier of R^*n by VECTSP_4:def 2;
   now let o be object;
   assume o in the carrier of R^*n; then
   consider l being Linear_Combination of Base(R,n) such that
   C: Sum l = o by lemspan1;
   thus o in the carrier of V by C,A;
   end; then
the carrier of R^*n c= the carrier of V;
hence thesis by B,XBOOLE_0:def 10,VECTSP_4:31;
end;
