let V be RealUnitarySpace; :: thesis: (0). V is finite-dimensional

reconsider V9 = (0). V as strict RealUnitarySpace ;

reconsider I = {} the carrier of V9 as finite Subset of V9 ;

the carrier of V9 = {(0. V)} by RUSUB_1:def 2

.= {(0. V9)} by RUSUB_1:4

.= the carrier of ((0). V9) by RUSUB_1:def 2 ;

then A1: V9 = (0). V9 by RUSUB_1:26;

( I is linearly-independent & Lin I = (0). V9 ) by RLVECT_3:7, RUSUB_3:3;

then I is Basis of V9 by A1, RUSUB_3:def 2;

hence (0). V is finite-dimensional ; :: thesis: verum

reconsider V9 = (0). V as strict RealUnitarySpace ;

reconsider I = {} the carrier of V9 as finite Subset of V9 ;

the carrier of V9 = {(0. V)} by RUSUB_1:def 2

.= {(0. V9)} by RUSUB_1:4

.= the carrier of ((0). V9) by RUSUB_1:def 2 ;

then A1: V9 = (0). V9 by RUSUB_1:26;

( I is linearly-independent & Lin I = (0). V9 ) by RLVECT_3:7, RUSUB_3:3;

then I is Basis of V9 by A1, RUSUB_3:def 2;

hence (0). V is finite-dimensional ; :: thesis: verum