reserve V for RealLinearSpace,
  W for Subspace of V,
  x, y, y1, y2 for set,
  i, n for Element of NAT,
  v for VECTOR of V,
  KL1, KL2 for Linear_Combination of V,
  X for Subset of V;
reserve V for finite-dimensional RealLinearSpace,
  W, W1, W2 for Subspace of V,
  u, v for VECTOR of V;

theorem
  dim V = dim W iff (Omega).V = (Omega).W
proof
  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 Th16;
    the carrier of W c= the carrier of V by RLSUB_1:def 2;
    then reconsider A9= A as finite Subset of V by Th23,XBOOLE_1:1;
    reconsider B9= B as finite Subset of V by Th23;
    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,XBOOLE_0:def 8;
      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 RLSStruct of V by RLSUB_1:def 4
      .= Lin(B) by RLVECT_3:def 3
      .= Lin(A) by A4,Th20
      .= the RLSStruct of W by RLVECT_3:def 3
      .= (Omega).W by RLSUB_1:def 4;
    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,RLVECT_3:def 3;
  assume (Omega).V = (Omega).W;
  then the RLSStruct of V = (Omega).W by RLSUB_1:def 4
    .= the RLSStruct of W by RLSUB_1:def 4;
  then
A7: Lin(A) = the RLSStruct of W by A1,RLVECT_3:def 3
    .= Lin(B) by A5,RLVECT_3:def 3;
A8: B is linearly-independent by A5,RLVECT_3:def 3;
  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,Th29
    .= card B by A8,Th29
    .= dim W by A5,Def2;
  hence thesis;
end;
