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
reconsider V' = V as VectSp of GF ;
consider I being Basis of V';
A1: Lin I = VectSpStr(# the carrier of V',the U5 of V',the U2 of V',the lmult of V' #) by VECTSP_7:def 3;
reconsider I = I as finite Subset of V by Th24;
consider A being Basis of W;
reconsider A = A as Subset of W ;
A is linearly-independent by VECTSP_7:def 3;
then reconsider B = A as linearly-independent Subset of V by Th15;
reconsider A' = B as finite Subset of V by Th24;
A2: dim W = card A by Def2;
card A' <= card I by A1, Th23;
hence dim W <= dim V by A2, Def2; :: thesis: verum