
theorem Th9:
for n,b being Nat st b > 1 holds n < b|^(len digits(n,b))
proof
  let n,b be Nat;
  assume A1: b > 1;
  per cases;
  suppose n=0;
    hence thesis by A1;
  end;
  suppose n<>0;
    then A2: value(digits(n,b),b)=n &
    digits(n,b).(len(digits(n,b))-1) <> 0 &
    for i being Nat st i in dom digits(n,b) holds 0 <= digits(n,b).i &
    digits(n,b).i < b by NUMERAL1:def 2,A1;
    A3: len digits(n,b) > 0 by NUMERAL1:4,A1;
    for i being Nat st i in dom digits(n,b) holds digits(n,b).i<b by A2;
    then value(digits(n,b),b) < b|^(len digits(n,b)) by A1,A3,Th8;
    hence n < b|^(len digits(n,b)) by A2;
  end;
end;
