
theorem Th5:
for b being Nat st b > 1
for s being NAT-valued XFinSequence st len s > 0 & s.((len s)-1) <> 0 &
for i being Nat st i in dom s holds s.i<b holds digits(value(s,b),b) = s
proof
  let b be Nat;
  assume A1: b > 1;
  let s be NAT-valued XFinSequence;
  assume A2: len s > 0 & s.((len s)-1) <> 0 &
  for i being Nat st i in dom s holds s.i<b;
  reconsider l=(len s)-1 as Nat by A2,NAT_1:20;
  A3: b|^l <> 0 by A1;
  consider d9 being XFinSequence of NAT such that
  A4: (dom d9 = dom s &
  for i being Nat st i in dom d9 holds d9.i = (s.i)*(b|^i)) &
  value(s,b) = Sum d9 by NUMERAL1:def 1;
  A5: len d9 <> 0 by A2,A4;
  l in dom d9 by A2,A4,FUNCT_1:def 2;
  then d9.l=(s.l)*(b|^l) by A4;
  then d9.l<>0 by A2,A3,XCMPLX_1:6;
  then ((len d9) --> 0).l <> d9.l;
  then A6: value(s,b)<>0 by A4,AFINSQ_2:62,A5;
  for i being Nat st i in dom s holds s.i>=0 & s.i<b by A2;
  hence digits(value(s,b),b) = s by A6,A1,A2,NUMERAL1:def 2;
end;
