reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem
  for X be finite-dimensional RealLinearSpace,
      b be OrdBasis of RLSp2RVSp(X),
      x, y be Element of X,
      a be Real
  holds
    0 <= (euclid_norm(X,b)).x
    & ((euclid_norm(X,b)).x = 0 iff x = 0.X)
    & (euclid_norm(X,b)).(a*x) = |.a.| * (euclid_norm(X,b)).x
    & (euclid_norm(X,b)).(x+y) <= (euclid_norm(X,b)).x + (euclid_norm(X,b)).y
  proof
    let X be finite-dimensional RealLinearSpace,
        b be OrdBasis of RLSp2RVSp(X),
        x, y be Element of X,
        a be Real;

    set xSUM = (euclid_norm(X,b)).x;
    set ySUM = (euclid_norm(X,b)).y;
    set axSUM = (euclid_norm(X,b)).(a*x);
    set xySUM = (euclid_norm(X,b)).(x+y);

    consider x1 be Element of RLSp2RVSp(X),
             z1 be Element of REAL(dim X) such that
    A1: x = x1
      & z1 = (x1 |-- b)
      & xSUM = |.z1.| by Def5;

    consider x2 be Element of RLSp2RVSp(X),
             z2 be Element of REAL(dim X) such that
    A2: y = x2
      & z2 = (x2 |-- b)
      & ySUM = |.z2.| by Def5;

    consider xy be Element of RLSp2RVSp(X),
             z3 be Element of REAL(dim X) such that
    A3: x + y = xy
      & z3 = (xy |-- b)
      & xySUM = |.z3.| by Def5;

    consider ax be Element of RLSp2RVSp(X),
             z4 be Element of REAL(dim X) such that
    A4: a * x = ax
      & z4 = (ax |-- b)
      & axSUM = |.z4.| by Def5;

    thus 0 <= xSUM by A1;

    0* (dim X)
     = (dim (RLSp2RVSp (X))) |-> (0. F_Real) by REAL_NS2:81
    .= (len (b)) |-> (0. F_Real) by MATRLIN2:21
    .= (0. RLSp2RVSp (X)) |-- b by MATRLIN2:20;

    hence xSUM = 0 iff x = 0.X by A1,EUCLID:7,EUCLID:8,MATRLIN:34;
    reconsider a1 = a as Element of F_Real by XREAL_0:def 1;
    ax = a1 * x1 by A1,A4; then
    z4 = a1 * (x1 |-- b) by A4,MATRLIN2:18
    .= a * z1 by A1;

    hence axSUM = |.a.| * xSUM by A1,A4,EUCLID:11;

    xy = x1 + x2 by A1,A2,A3;
    then
    z3 = (x1 |-- b) + (x2 |-- b) by A3,MATRLIN2:17
      .= z1 + z2 by A1,A2;
    hence xySUM <= xSUM + ySUM by A1,A2,A3,EUCLID:12;
  end;
