let V be finite-dimensional RealLinearSpace; :: 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 = RLSStruct(# the carrier of V, the ZeroF of V, the U5 of V, the Mult of V #) by RLSUB_1:def 4
.= Lin I by A2, RLVECT_3:def 3
.= (0). V by A3, RLVECT_3:16 ;
hence (Omega). V = (0). V ; :: thesis: verum
end;
A4: I <> {(0. V)} by A1, RLVECT_3:8, RLVECT_3:def 3;
assume (Omega). V = (0). V ; :: thesis: dim V = 0
then RLSStruct(# the carrier of V, the ZeroF of V, the U5 of V, the Mult of V #) = (0). V by RLSUB_1:def 4;
then Lin I = (0). V by A1, RLVECT_3:def 3;
then ( I = {} or I = {(0. V)} ) by RLVECT_3:17;
hence dim V = 0 by A1, A4, Def2, CARD_1:27; :: thesis: verum