let V be RealLinearSpace; :: thesis: ( V is finite-dimensional implies for A, B being Basis of V holds card A = card B )
assume A1: V is finite-dimensional ; :: thesis: for A, B being Basis of V holds card A = card B
let A, B be Basis of V; :: thesis: card A = card B
reconsider A' = A, B' = B as finite Subset of V by A1, Th24;
A2: RLSStruct(# the carrier of V,the U2 of V,the U5 of V,the Mult of V #) = Lin A by RLVECT_3:def 3;
B' is linearly-independent by RLVECT_3:def 3;
then A3: card B' <= card A' by A2, Th23;
A4: RLSStruct(# the carrier of V,the U2 of V,the U5 of V,the Mult of V #) = Lin B by RLVECT_3:def 3;
A' is linearly-independent by RLVECT_3:def 3;
then card A' <= card B' by A4, Th23;
hence card A = card B by A3, XXREAL_0:1; :: thesis: verum