let V be finite-dimensional RealLinearSpace; :: thesis: for W being Subspace of V holds dim W <= dim V
let W be Subspace of V; :: thesis: dim W <= dim V
reconsider V9 = V as RealLinearSpace ;
set I = the Basis of V9;
reconsider I = the Basis of V9 as finite Subset of V by Th24;
set A = the Basis of W;
reconsider A = the Basis of W as Subset of W ;
A1: A is linearly-independent by RLVECT_3:def 3;
then reconsider A9 = A as finite Subset of V by Th15, Th24;
( Lin I = RLSStruct(# the carrier of V9, the ZeroF of V9, the U5 of V9, the Mult of V9 #) & A is linearly-independent Subset of V ) by A1, Th15, RLVECT_3:def 3;
then A2: card A9 <= card I by Th23;
dim W = card A by Def3;
hence dim W <= dim V by A2, Def3; :: thesis: verum