
theorem
  for V being finite-dimensional RealUnitarySpace, W1,W2 being Subspace
  of V holds dim(W1 /\ W2) >= dim W1 + dim W2 - dim V
proof
  let V be finite-dimensional RealUnitarySpace;
  let W1,W2 be Subspace of V;
A1: dim(W1 + W2) <= dim V & dim V + (dim(W1 /\ W2) - dim V) = dim(W1 /\ W2)
  by Th8;
  dim W1 + dim W2 - dim V = dim(W1 + W2) + dim(W1 /\ W2) - dim V by Th15
    .= dim(W1 + W2) + (dim(W1 /\ W2) - dim V);
  hence thesis by A1,XREAL_1:6;
end;
