let GF be Field; :: thesis: for V being finite-dimensional VectSp of GF
for A being Subset of V st A is linearly-independent holds
card A = dim (Lin A)
let V be finite-dimensional VectSp of GF; :: thesis: for A being Subset of V st A is linearly-independent holds
card A = dim (Lin A)
let A be Subset of V; :: thesis: ( A is linearly-independent implies card A = dim (Lin A) )
assume A1: A is linearly-independent ; :: thesis: card A = dim (Lin A)
set W = Lin A;
for x being object st x in A holds
x in the carrier of (Lin A) by STRUCT_0:def 5, VECTSP_7:8;
then reconsider B = A as linearly-independent Subset of (Lin A) by A1, Th12, TARSKI:def 3;
Lin A = Lin B by Th17;
then reconsider B = B as Basis of (Lin A) by VECTSP_7:def 3;
card B = dim (Lin A) by Def1;
hence card A = dim (Lin A) ; :: thesis: verum