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
  V is_the_direct_sum_of W1, W2 implies dim V = dim W1 + dim W2
proof
  assume
A1: V is_the_direct_sum_of W1, W2;
  then
A2: the RLSStruct of V = W1 + W2 by RLSUB_2:def 4;
  W1 /\ W2 = (0).V by A1,RLSUB_2:def 4;
  then (Omega).(W1 /\ W2) = (0).V by RLSUB_1:def 4
    .= (0).(W1 /\ W2) by RLSUB_1:36;
  then dim(W1 /\ W2) = 0 by Th32;
  then dim W1 + dim W2 = dim(W1 + W2) + 0 by Th35
    .= dim (Omega).V by A2,RLSUB_1:def 4
    .= dim V by Th30;
  hence thesis;
end;
