let GF be Field; :: thesis: for V being finite-dimensional VectSp of GF
for W being Subspace of V holds dim W <= dim V
let V be finite-dimensional VectSp of GF; :: thesis: for W being Subspace of V holds dim W <= dim V
let W be Subspace of V; :: thesis: dim W <= dim V
consider A being Basis of W;
reconsider A = A as Subset of W ;
A1: dim W = card A by Def2;
A is linearly-independent by VECTSP_7:def 3;
then reconsider B = A as linearly-independent Subset of V by Th15;
reconsider A9 = B as finite Subset of V by Th24;
reconsider V9 = V as VectSp of GF ;
consider I being Basis of V9;
A2: Lin I = VectSpStr(# the carrier of V9, the U5 of V9, the ZeroF of V9, the lmult of V9 #) by VECTSP_7:def 3;
reconsider I = I as finite Subset of V by Th24;
card A9 <= card I by A2, Th23;
hence dim W <= dim V by A1, Def2; :: thesis: verum