let V be RealUnitarySpace; :: thesis: (0). V is finite-dimensional
reconsider V' = (0). V as strict RealUnitarySpace ;
the carrier of V' = {(0. V)} by RUSUB_1:def 2
.= {(0. V')} by RUSUB_1:4
.= the carrier of ((0). V') by RUSUB_1:def 2 ;
then A1: V' = (0). V' by RUSUB_1:26;
reconsider I = {} the carrier of V' as finite Subset of V' ;
A2: I is linearly-independent by RLVECT_3:8;
Lin I = (0). V' by RUSUB_3:3;
then I is Basis of V' by A1, A2, RUSUB_3:def 2;
hence (0). V is finite-dimensional by Def1; :: thesis: verum