reserve V for RealLinearSpace;

theorem Th23:
  for V being RealNormSpace holds the normF of V is
  absolutely_homogeneous subadditive Functional of V
proof
  let V be RealNormSpace;
  reconsider N = the normF of V as Functional of V;
A1: N is subadditive
  proof
    let x,y be VECTOR of V;
A2: N.(x+y) = ||.x+y.|| by NORMSP_0:def 1;
    N.(x) = ||.x.|| & N.(y) = ||.y.|| by NORMSP_0:def 1;
    hence thesis by A2,NORMSP_1:def 1;
  end;
  N is absolutely_homogeneous
  proof
    let x be VECTOR of V, r be Real;
    thus N.(r*x) = ||.r*x.|| by NORMSP_0:def 1
      .= |.r.|*||.x.|| by NORMSP_1:def 1
      .= |.r.|*N.x by NORMSP_0:def 1;
  end;
  hence thesis by A1;
end;
