let V be finite-dimensional RealUnitarySpace; :: thesis: for n being Element of NAT st n <= dim V holds
not n Subspaces_of V is empty
let n be Element of NAT ; :: thesis: ( n <= dim V implies not n Subspaces_of V is empty )
assume n <= dim V ; :: thesis: not n Subspaces_of V is empty
then ex W being strict Subspace of V st dim W = n by Lm2;
hence not n Subspaces_of V is empty by Def3; :: thesis: verum