
theorem Th708:
digits(3800,10) = <%0,0,8,3%>
proof
  set d=<%0,0,8,3%>;
  set e=<%0*10|^0,0*10|^1,8*10|^2,3*10|^3%>;
  A1: Sum e = Sum (<%0*10|^0,0*10|^1,8*10|^2%>)+Sum(<%3*10|^3%>) by AFINSQ_2:55
  .= Sum (<%0*10|^0,0*10|^1%>)+Sum(<%8*10|^2%>)+Sum(<%3*10|^3%>) by AFINSQ_2:55
  .= ((0*10|^0)+(0*10|^1))+Sum(<%8*10|^2%>)+Sum(<%3*10|^3%>) by AFINSQ_2:54
  .= ((0*10|^0)+(0*10|^1))+(8*10|^2)+Sum(<%3*10|^3%>) by AFINSQ_2:53
  .= 0*1 + 0*(10|^1) + 8*(10|^2) + 3*(10|^3) by AFINSQ_2:53
  .= 0 + 8*(10*10) + 3*(10|^3) by POLYEQ_5:1
  .= 0 + 8*(10*10) + 3*(10*10*10) by POLYEQ_5:2
  .= 3800;
  A2: dom d = 4 by AFINSQ_1:84 .= dom e by AFINSQ_1:84;
  now
    let i be Nat;
    assume i in dom d;
    then i in 4 by AFINSQ_1:84;
    then i in {0,1,2,3} by CARD_1:52;
    then i = 0 or i = 1 or i = 2 or i = 3 by ENUMSET1:def 2;
    hence e.i=(d.i)*10|^i;
  end;
  then A3: value(d,10)=3800 by A1,A2,NUMERAL1:def 1;
  len(d) - 1 = 4-1 by AFINSQ_1:84;
  then A4: d.(len(d)-1) <> 0;
  now
    let i be Nat;
    assume i in dom d;
    then i in 4 by AFINSQ_1:84;
    then i in {0,1,2,3} by CARD_1:52;
    then i = 0 or i = 1 or i = 2 or i = 3 by ENUMSET1:def 2;
    hence 0 <= d.i & d.i < 10;
  end;
  hence thesis by A3,A4,NUMERAL1:def 2;
end;
