
theorem
  for n be Nat st n >= 1 holds for m,k be Nat st k >= 2 holds SDDec(
  SDMax(n,m,k)) + SDDec(SDMin(n,m,k)) = SDDec(DecSD(0,n,k))
proof
  let n be Nat;
  assume
A1: n >= 1;
  then
A2: n in Seg n by FINSEQ_1:1;
  let m,k be Nat;
  assume
A3: k >= 2;
A4: for i be Nat st i in Seg n holds DigA(DecSD(0,n,k),i) = DigA(SDMax(n,m,k
  ) '+' SDMin(n,m,k),i)
  proof
    let i be Nat;
    assume
A5: i in Seg n;
    then
A6: DigA(SDMax(n,m,k) '+' SDMin(n,m,k),i) = Add(SDMax(n,m,k),SDMin(n,m,k),
    i,k) by RADIX_1:def 14
      .= SD_Add_Data(DigA(SDMax(n,m,k),i)+DigA(SDMin(n,m,k),i),k) +
SD_Add_Carry(DigA(SDMax(n,m,k),i-'1)+DigA(SDMin(n,m,k),i-'1)) by A3,A5,
RADIX_1:def 13
      .= SD_Add_Data(0,k) + SD_Add_Carry(DigA(SDMax(n,m,k),i-'1)+DigA(SDMin(
    n,m,k),i-'1)) by A3,A5,Th16
      .= 0+SD_Add_Carry(DigA(SDMax(n,m,k),i-'1)+DigA(SDMin(n,m,k),i-'1)) by
RADIX_1:19;
A7: DigA(DecSD(0,n,k),i) = 0 by A5,Th5;
A8: i >= 1 by A5,FINSEQ_1:1;
    now
      per cases by A8,XXREAL_0:1;
      suppose
A9:     i = 1;
        then DigA(SDMin(n,m,k),i-'1) = 0 by NAT_2:8,RADIX_1:def 3;
        then DigA(SDMax(n,m,k) '+' SDMin(n,m,k),i) = SD_Add_Carry(0+0) by A6,A9
,NAT_2:8,RADIX_1:def 3;
        hence thesis by A7,RADIX_1:def 10;
      end;
      suppose
        i > 1;
        then i -' 1 in Seg n by A5,Th2;
        then
        DigA(SDMax(n,m,k) '+' SDMin(n,m,k),i) = SD_Add_Carry(0) by A3,A6,Th16;
        hence thesis by A7,RADIX_1:def 10;
      end;
    end;
    hence thesis;
  end;
  SDDec(SDMax(n,m,k)) + SDDec(SDMin(n,m,k)) = SDDec(SDMax(n,m,k) '+'
  SDMin(n,m,k)) + SD_Add_Carry(DigA(SDMax(n,m,k),n)+DigA(SDMin(n,m,k),n)) * (
  Radix(k) |^ n) by A1,A3,RADIX_2:11
    .= SDDec(SDMax(n,m,k) '+' SDMin(n,m,k)) + SD_Add_Carry(0) * (Radix(k) |^
  n) by A3,A2,Th16
    .= SDDec(SDMax(n,m,k) '+' SDMin(n,m,k)) + 0 * (Radix(k) |^ n) by
RADIX_1:def 10;
  hence thesis by A1,A4,Th12;
end;
