let V be RealUnitarySpace; :: thesis: for A being Subset of V st V is finite-dimensional & A is linearly-independent holds
A is finite
let A be Subset of V; :: thesis: ( V is finite-dimensional & A is linearly-independent implies A is finite )
assume that
A1: V is finite-dimensional and
A2: A is linearly-independent ; :: thesis: A is finite
consider B being Basis of V such that
A3: A c= B by A2, RUSUB_3:15;
B is finite by A1, Th3;
hence A is finite by A3; :: thesis: verum