
theorem Th780:
digits(704,10) = <%4,0,7%>
proof
  set d=<%4,0,7%>;
  set e=<%4*10|^0,0*10|^1,7*10|^2%>;
  A1: Sum e = Sum (<%4*10|^0,0*10|^1%>)+Sum(<%7*10|^2%>) by AFINSQ_2:55
  .= ((4*10|^0)+(0*10|^1))+Sum(<%7*10|^2%>) by AFINSQ_2:54
  .= ((4*10|^0)+(0*10|^1))+(7*10|^2) by AFINSQ_2:53
  .= 4*1 + 0*(10|^1) + 7*(10|^2) by NEWTON:4
  .= 4 + 7*(10*10) by POLYEQ_5:1
  .= 704;
  A2: dom d = 3 by AFINSQ_1:39 .= dom e by AFINSQ_1:39;
  now
    let i be Nat;
    assume i in dom d;
    then i in 3 by AFINSQ_1:39;
    then i in {0,1,2} by CARD_1:51;
    then i = 0 or i = 1 or i = 2 by ENUMSET1:def 1;
    hence e.i=(d.i)*10|^i;
  end;
  then A3: value(d,10)=704 by A1,A2,NUMERAL1:def 1;
  len(d) - 1 = 3-1 by AFINSQ_1:39;
  then A4: d.(len(d)-1) <> 0;
  now
    let i be Nat;
    assume i in dom d;
    then i in 3 by AFINSQ_1:39;
    then i in {0,1,2} by CARD_1:51;
    then i = 0 or i = 1 or i = 2 by ENUMSET1:def 1;
    hence 0 <= d.i & d.i < 10;
  end;
  hence thesis by A3,A4,NUMERAL1:def 2;
end;
