
theorem Th3:
  for m,k be Nat st k >= 2 holds for r be Tuple of (m+2),k-SD holds
  SDDec(Mmax(r)) >= SDDec(r)
proof
  let m,k be Nat;
  assume that
A1: k >= 2;
  let r be Tuple of (m+2),k-SD;
  for i be Nat st i in Seg (m+2) holds DigA(Mmax(r),i) >= DigA(r,i)
  proof
    let i be Nat;
    assume
A2: i in Seg (m+2);
    then
A3: DigA(Mmax(r),i) = MmaxDigit(r,i) by Def4;
    now
      per cases;
      suppose
        i >= m;
        then DigA(Mmax(r),i) = r.i by A1,A2,A3,Def3
          .= DigA(r,i) by A2,RADIX_1:def 3;
        hence thesis;
      end;
      suppose
A4:     i < m;
A5:     DigA(r,i) = r.i by A2,RADIX_1:def 3;
A6:     r.i is Element of k-SD by A2,RADIX_1:15;
        DigA(Mmax(r),i) = Radix(k) - 1 by A1,A2,A3,A4,Def3;
        hence thesis by A5,A6,RADIX_1:13;
      end;
    end;
    hence thesis;
  end;
  hence thesis by NAT_1:12,RADIX_5:13;
end;
