
theorem Th12:
  for V being finite-dimensional RealUnitarySpace holds dim V = 0
  iff (Omega).V = (0).V
proof
  let V be finite-dimensional RealUnitarySpace;
  consider I being finite Subset of V such that
A1: I is Basis of V by Def1;
  hereby
    consider I being finite Subset of V such that
A2: I is Basis of V by Def1;
    assume dim V = 0;
    then card I = 0 by A2,Def2;
    then
A3: I = {}(the carrier of V);
    (Omega).V = the UNITSTR 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;
  end;
A4: I <> {0.V} by A1,RUSUB_3:def 2,RLVECT_3:8;
  assume (Omega).V = (0).V;
  then the UNITSTR 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 thesis by A1,A4,Def2,CARD_1:27;
end;
