
theorem Th8:
  for V being finite-dimensional RealUnitarySpace, W being Subspace
  of V holds dim W <= dim V
proof
  let V be finite-dimensional RealUnitarySpace;
  let W be Subspace of V;
  set A = the Basis of W;
  reconsider A as Subset of W;
  A is linearly-independent by RUSUB_3:def 2;
  then reconsider B=A as linearly-independent Subset of V by RUSUB_3:22;
  reconsider A9= B as finite Subset of V by Th3;
  reconsider V9= V as RealUnitarySpace;
  set I = the Basis of V9;
A1: Lin(I) = the UNITSTR of V9 by RUSUB_3:def 2;
  reconsider I as finite Subset of V by Th3;
A2: dim V = card I by Def2;
  card A9 <= card I by A1,Th2;
  hence thesis by A2,Def2;
end;
