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 Th19:
  for X be finite-dimensional RealLinearSpace,
      b be OrdBasis of RLSp2RVSp(X),
      x,y be Element of X,
      a be Real
  st dim X <> 0
  holds
    0 <= (max_norm(X,b)).x
    & ((max_norm(X,b)).x = 0 iff x = 0.X)
    & (max_norm(X,b)).(a*x) = |.a.| * (max_norm(X,b)).x
    & (max_norm(X,b)).(x+y) <= (max_norm(X,b)).x + (max_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;
    assume
    A1: dim X <> 0;

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

    consider x1 be Element of RLSp2RVSp (X),
             z1 be Element of REAL (dim X) such that
    A2: x = x1
      & z1 = x1 |-- b
      & xSUM = (max_norm(dim X)).z1 by Def3;

    consider x2 be Element of RLSp2RVSp(X),
             z2 be Element of REAL(dim X) such that
    A3: y = x2
      & z2 = x2 |-- b
      & ySUM = (max_norm(dim X)).z2 by Def3;

    consider xy be Element of RLSp2RVSp(X),
             z3 be Element of REAL(dim X) such that
    A4: x + y = xy
      & z3 = xy |-- b
      & xySUM = (max_norm(dim X)).z3 by Def3;

    consider ax be Element of RLSp2RVSp(X),
             z4 be Element of REAL(dim X) such that
    A5: a*x = ax
      & z4 = ax |-- b
      & axSUM = (max_norm(dim X)).z4 by Def3;

    thus 0 <= xSUM by A1,A2,Th12;

    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,A2,Th12,MATRLIN:34;
    reconsider a1 = a as Element of F_Real by XREAL_0:def 1;
    ax = a1 * x1 by A2,A5; then
    z4 = a1 * (x1 |-- b) by A5,MATRLIN2:18
          .= a * z1 by A2;

    hence axSUM = |.a.| * xSUM by A1,A2,A5,Th12;

    xy = x1 + x2 by A2,A3,A4; then
    z3 = (x1 |-- b) + (x2 |-- b) by A4,MATRLIN2:17
      .= z1 + z2 by A2,A3;
    hence xySUM <= xSUM+ySUM by A1,A2,A3,A4,Th12;
  end;
