
theorem Th5:
  for V being RealUnitarySpace, A,B being Basis of V st V is
  finite-dimensional holds card A = card B
proof
  let V be RealUnitarySpace;
  let A, B be Basis of V;
  assume V is finite-dimensional;
  then reconsider A9= A, B9= B as finite Subset of V by Th3;
  the UNITSTR of V = Lin(B) & A9 is linearly-independent by RUSUB_3:def 2;
  then
A1: card A9 <= card B9 by Th2;
  the UNITSTR of V = Lin(A) & B9 is linearly-independent by RUSUB_3:def 2;
  then card B9 <= card A9 by Th2;
  hence thesis by A1,XXREAL_0:1;
end;
