let V be finite-dimensional RealUnitarySpace; :: thesis: ( dim V = 0 iff (Omega). V = (0). V )
consider I being finite Subset of V such that
A1: I is Basis of V by Def1;
hereby :: thesis: ( (Omega). V = (0). V implies dim V = 0 )
consider I being finite Subset of V such that
A2: I is Basis of V by Def1;
assume dim V = 0 ; :: thesis: (Omega). V = (0). V
then card I = 0 by A2, Def2;
then A3: I = {} the carrier of V ;
(Omega). V = UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) by RUSUB_1:def 3
.= Lin I by A2, RUSUB_3:def 2
.= (0). V by A3, RUSUB_3:3 ;
hence (Omega). V = (0). V ; :: thesis: verum
end;
A4: I <> {(0. V)} by A1, RUSUB_3:def 2, RLVECT_3:8;
assume (Omega). V = (0). V ; :: thesis: dim V = 0
then UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = (0). V by RUSUB_1:def 3;
then Lin I = (0). V by A1, RUSUB_3:def 2;
then ( I = {} or I = {(0. V)} ) by RUSUB_3:4;
hence dim V = 0 by A1, A4, Def2, CARD_1:27; :: thesis: verum