
theorem
  for V being finite-dimensional RealUnitarySpace, W being Subspace of V
  holds dim V = dim W iff (Omega).V = (Omega).W
proof
  let V be finite-dimensional RealUnitarySpace;
  let W be Subspace of V;
  consider A being finite Subset of V such that
A1: A is Basis of V by Def1;
  hereby
    set A = the Basis of W;
    consider B being Basis of V such that
A2: A c= B by RUSUB_3:24;
    the carrier of W c= the carrier of V by RUSUB_1:def 1;
    then reconsider A9= A as finite Subset of V by Th3,XBOOLE_1:1;
    reconsider B9= B as finite Subset of V by Th3;
    assume dim V = dim W;
    then
A3: card A = dim V by Def2
      .= card B by Def2;
A4: now
      assume A <> B;
      then A c< B by A2;
      then card A9 < card B9 by CARD_2:48;
      hence contradiction by A3;
    end;
    reconsider B as Subset of V;
    reconsider A as Subset of W;
    (Omega).V = the UNITSTR of V by RUSUB_1:def 3
      .= Lin(B) by RUSUB_3:def 2
      .= Lin(A) by A4,RUSUB_3:28
      .= the UNITSTR of W by RUSUB_3:def 2
      .= (Omega).W by RUSUB_1:def 3;
    hence (Omega).V = (Omega).W;
  end;
  consider B being finite Subset of W such that
A5: B is Basis of W by Def1;
A6: A is linearly-independent by A1,RUSUB_3:def 2;
  assume (Omega).V = (Omega).W;
  then the UNITSTR of V = (Omega).W by RUSUB_1:def 3
    .= the UNITSTR of W by RUSUB_1:def 3;
  then
A7: Lin(A) = the UNITSTR of W by A1,RUSUB_3:def 2
    .= Lin(B) by A5,RUSUB_3:def 2;
A8: B is linearly-independent by A5,RUSUB_3:def 2;
  reconsider B as Subset of W;
  reconsider A as Subset of V;
  dim V = card A by A1,Def2
    .= dim Lin(B) by A6,A7,Th9
    .= card B by A8,Th9
    .= dim W by A5,Def2;
  hence thesis;
end;
