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 Th12:
  for n be non empty Nat,
      x,y be Element of REAL n,
      a be Real
  holds
      0 <= (max_norm(n)).x
    & ((max_norm(n)).x = 0 iff x = 0*n)
    & (max_norm(n)).(a * x) = |.a.| * (max_norm(n)).x
    & (max_norm(n)).(x + y) <= (max_norm(n)).x + (max_norm(n)).y
  proof
    let n be non empty Nat,
        x, y be Element of REAL n,
        a be Real;

    set xMAX = (max_norm(n)).x;
    set yMAX = (max_norm(n)).y;
    set axMAX = (max_norm(n)).(a*x);
    set xyMAX = (max_norm(n)).(x+y);

    A1: xMAX in rng (abs x)
      & for i be Nat st i in dom x
        holds abs(x).i <= xMAX by Def1;

    A2: axMAX in rng (abs(a*x))
      & for i be Nat st i in dom (a*x)
        holds abs(a*x).i <= axMAX by Def1;

    A3: xyMAX in rng (abs(x+y))
      & for i be Nat st i in dom (x+y)
        holds abs(x+y).i <= xyMAX by Def1;

    A4: dom x = dom (abs x) by VALUED_1:def 11;
    A5: dom y = dom (abs y) by VALUED_1:def 11;
    A6: len x = n by CARD_1:def 7;
    then
    A7: dom x = Seg n by FINSEQ_1:def 3;

    len y = n by CARD_1:def 7;
    then
    A8: dom y = Seg n by FINSEQ_1:def 3;
    len (x+y) = n by CARD_1:def 7;
    then
    A9: dom (x+y) = Seg n by FINSEQ_1:def 3;

    xMAX in rng (abs x) by Def1;
    then consider j0 be object such that
    A10: j0 in dom (abs x) & xMAX = (abs x).j0 by FUNCT_1:def 3;
    reconsider j0 as Nat by A10;
    (abs x).j0 = |. x.j0 .| by A10,VALUED_1:def 11;
    hence 0 <= (max_norm(n)).x by A10;

    yMAX in rng (abs y) by Def1;
    then consider k0 be object such that
    A11: k0 in dom (abs y) & yMAX =(abs y).k0 by FUNCT_1:def 3;
    reconsider k0 as Nat by A11;

    thus (max_norm(n)).x = 0 iff x=0*n
    proof
      hereby
        assume
        A12: (max_norm(n)).x = 0;

        for i be object st i in dom x holds x.i = 0
        proof
          let i0 be object;
          assume
          A13: i0 in dom x;
          then reconsider i = i0 as Nat;
          abs(x).i <= 0 by A12,A13,Def1;
          then |.x.i.| <= 0 by A4,A13,VALUED_1:def 11;
          hence x.i0 = 0;
        end;
        hence
        x = (dom x) --> 0 by FUNCOP_1:11
        .= (Seg n) --> (In (0,REAL)) by A6,FINSEQ_1:def 3
        .= 0* n by FINSEQ_2:def 2;
      end;
      assume
      A14: x = 0*n;
      consider j be object such that
      A15: j in dom (abs x)
         & xMAX = (abs x).j by A1,FUNCT_1:def 3;

      reconsider j as Nat by A15;
      (abs x).j
       = |. x.j .| by A15,VALUED_1:def 11
      .= 0 by A14;
      hence xMAX = 0 by A15;
    end;

    thus
    (max_norm(n)).(a*x) = |.a.| * (max_norm(n)).x
    proof
      set L = |.a.| * (max_norm(n)).x;
      A16: dom(abs(a*x)) = dom(a*x) by VALUED_1:def 11;
      A17: dom(a*x) = dom x by VALUED_1:def 5;
      A18: j0 in dom(a*x) by A4,A10,VALUED_1:def 5;
      then
      (abs (a*x)).j0
       = |. (a*x).j0 .| by A16,VALUED_1:def 11
      .= |. a*(x.j0) .| by A18,VALUED_1:def 5
      .= |.a.| * |.x.j0.| by COMPLEX1:65
      .= L by A10,VALUED_1:def 11;
      then
      A19: L <= (max_norm(n)).(a*x) by Def1,A18;
      consider k0 be object such that
      A20: k0 in dom (abs (a*x))
         & axMAX =(abs (a*x)).k0 by FUNCT_1:def 3,A2;
      reconsider j0 as Nat;
      A21: (abs (a*x)).k0
       = |. (a*x).k0 .| by A20,VALUED_1:def 11
      .= |.a*(x.k0).| by A16,A20,VALUED_1:def 5
      .= |.a.| * |.x.k0.| by COMPLEX1:65;

      A22: abs(x).k0 <= xMAX by A16,A17,A20,Def1;
      abs(x).k0 = |. x.k0 .| by A4,A16,A17,A20,VALUED_1:def 11;
      then axMAX <= L by A20,A21,A22,XREAL_1:64;
      hence (max_norm(n)).(a*x) = L by A19,XXREAL_0:1;
    end;

    thus (max_norm(n)).(x+y) <= (max_norm(n)).x+(max_norm(n)).y
    proof
      A23: dom (abs(x+y)) = dom (x+y) by VALUED_1:def 11;

      consider h0 be object such that
      A24: h0 in dom (abs(x+y))
         & xyMAX = (abs (x+y)).h0 by A3,FUNCT_1:def 3;
      A25: h0 in Seg n by A9,A24,VALUED_1:def 11;

      reconsider j0 as Nat;

      A26: (abs (x+y)).h0
       = |. (x+y).h0 .| by A24,VALUED_1:def 11
      .= |.x.h0 + y.h0.| by A23,A24,VALUED_1:def 1;

      A27: |.x.h0 + y.h0.| <= |.x.h0.| + |.y.h0.| by COMPLEX1:56;
      abs(x).h0 <= xMAX by A7,A25,Def1;
      then
      A28: |. x.h0 .| <= xMAX by A4,A25,A7,VALUED_1:def 11;
      abs(y).h0 <= yMAX by A8,A25,Def1;
      then |. y.h0 .| <= yMAX by A5,A8,A25,VALUED_1:def 11;
      then |. x.h0 .| + |. y.h0 .| <= xMAX +yMAX by A28,XREAL_1:7;
      hence thesis by A24,A26,A27,XXREAL_0:2;
    end;
  end;
