let V be RealUnitarySpace; :: thesis: for A being Subset of V st A is linearly-independent holds
ex I being Basis of V st A c= I
let A be Subset of V; :: thesis: ( A is linearly-independent implies ex I being Basis of V st A c= I )
assume A is linearly-independent ; :: thesis: ex I being Basis of V st A c= I
then consider B being Subset of V such that
A1: A c= B and
A2: ( B is linearly-independent & Lin B = UNITSTR(# the carrier of V, the ZeroF of V, the U5 of V, the Mult of V, the scalar of V #) ) by Th11;
reconsider B = B as Basis of V by A2, Def2;
take B ; :: thesis: A c= B
thus A c= B by A1; :: thesis: verum