reserve i,n,m,k,x,y for Nat,
  i1 for Integer;

theorem Th7:
  for k,x,n be Nat st k >= 2 holds (Radix(k) |^ n) * DigA_SDSub(
SD2SDSub(DecSD(x,n+1,k)),n+1) = (Radix(k) |^ n) * DigA(DecSD(x,n+1,k),n+1) - (
Radix(k) |^ (n+1)) * SDSub_Add_Carry(DigA(DecSD(x,n+1,k),n+1),k) + (Radix(k) |^
  n) * SDSub_Add_Carry(DigA(DecSD(x,n+1,k),n),k)
proof
  let k,x,n be Nat;
  assume
A1: k >= 2;
A2: n+1 in Seg (n+1) by FINSEQ_1:3;
  then
A3: (n+1) in Seg (n+1+1) by FINSEQ_2:8;
  then (Radix(k) |^ n) * DigA_SDSub(SD2SDSub(DecSD(x,n+1,k)),n+1) = (Radix(k)
  |^ n) * (SD2SDSubDigitS(DecSD(x,n+1,k),n+1,k)) by RADIX_3:def 8
    .= (Radix(k) |^ n) * (SD2SDSubDigit(DecSD(x,n+1,k),n+1,k)) by A1,A3,
RADIX_3:def 7
    .= (Radix(k) |^ n) * ( SDSub_Add_Data(DigA(DecSD(x,n+1,k),n+1),k) +
  SDSub_Add_Carry(DigA(DecSD(x,n+1,k),n+1-'1),k)) by A2,RADIX_3:def 6
    .= (Radix(k) |^ n) * ( SDSub_Add_Data(DigA(DecSD(x,n+1,k),n+1),k) +
  SDSub_Add_Carry(DigA(DecSD(x,n+1,k),n),k) ) by NAT_D:34
    .= (Radix(k) |^ n) * ( DigA(DecSD(x,n+1,k),n+1) - Radix(k) *
SDSub_Add_Carry(DigA(DecSD(x,n+1,k),n+1),k) + SDSub_Add_Carry(DigA(DecSD(x,n+1,
  k),n),k) ) by RADIX_3:def 4
    .= (Radix(k) |^ n) * DigA(DecSD(x,n+1,k),n+1) - ((Radix(k) |^ n) * Radix
  (k)) * SDSub_Add_Carry(DigA(DecSD(x,n+1,k),n+1),k) + (Radix(k) |^ n) * (
  SDSub_Add_Carry(DigA(DecSD(x,n+1,k),n),k) )
    .= (Radix(k) |^ n) * DigA(DecSD(x,n+1,k),n+1) - (Radix(k) |^ (n+1)) *
  SDSub_Add_Carry(DigA(DecSD(x,n+1,k),n+1),k) + (Radix(k) |^ n) * (
  SDSub_Add_Carry(DigA(DecSD(x,n+1,k),n),k) ) by NEWTON:6;
  hence thesis;
end;
