
theorem Th229:
digits(85,10) = <%5,8%>
proof
  set d=<%5,8%>;
  set e=<%5*10|^0,8*10|^1%>;
  A1: Sum e = (5*10|^0)+(8*10|^1) by AFINSQ_2:54
  .= 5*1+(8*10|^1) by NEWTON:4
  .= 5 + 8*10 by NEWTON:5;
  A2: dom d = 2 by AFINSQ_1:38 .= dom e by AFINSQ_1:38;
  now
    let i be Nat;
    assume i in dom d;
    then i in 2 by AFINSQ_1:38;
    then i in {0,1} by CARD_1:50;
    then i = 0 or i = 1 by TARSKI:def 2;
    hence e.i=(d.i)*10|^i;
  end;
  then A3: value(d,10)=85 by A1,A2,NUMERAL1:def 1;
  len(d) - 1 = 2-1 by AFINSQ_1:38;
  then A4: d.(len(d)-1) <> 0;
  now
    let i be Nat;
    assume i in dom d;
    then i in 2 by AFINSQ_1:38;
    then i in {0,1} by CARD_1:50;
    then i = 0 or i = 1 by TARSKI:def 2;
    hence 0 <= d.i & d.i < 10;
  end;
  hence thesis by A3,A4,NUMERAL1:def 2;
end;
