
theorem Th26:
  for F being Field for V being finite-dimensional VectSp of F for
m,n being Nat st 1 <= m & m < n & n < dim V holds GrassmannSpace(V,m,n) is non
  degenerated
proof
  let F be Field;
  let V be finite-dimensional VectSp of F;
  let m,n be Nat such that
A1: 1 <= m and
A2: m < n and
A3: n < dim V;
  set S=GrassmannSpace(V,m,n);
A4: m < dim V by A2,A3,XXREAL_0:2;
  hereby
    assume
A5: the carrier of S is Block of S;
    the topology of S is non empty by A3,Th22;
    then consider W being Subspace of V such that
A6: dim W = n and
A7: m Subspaces_of V = m Subspaces_of W by A5,Def6;
    (Omega).V = (Omega).W by A1,A4,A7,Th24;
    then dim W = dim (Omega).V by VECTSP_9:27;
    hence contradiction by A3,A6,VECTSP_9:27;
  end;
end;
