
theorem Th16:
  for n,m,k be Nat st k >= 2 holds for i be Nat st i in Seg n
  holds DigA(SDMax(n,m,k),i)+DigA(SDMin(n,m,k),i) = 0
proof
  let n,m,k be Nat;
  assume
A1: k >= 2;
  let i be Nat;
  reconsider a = SDMinDigit(m,k,i) as Element of INT;
  reconsider b = SDMaxDigit(m,k,i) as Element of INT;
  assume
A2: i in Seg n;
  then
A3: i >= 1 by FINSEQ_1:1;
A4: DigA(SDMin(n,m,k),i) = SDMinDigit(m,k,i) by A2,Def2;
  now
    per cases;
    suppose
A5:   i < m;
      then a + b = -Radix(k) + 1 + b by A1,A3,Def1
        .= -Radix(k) + 1 + (Radix(k) - 1) by A1,A3,A5,Def3
        .= 0;
      hence thesis by A2,A4,Def4;
    end;
    suppose
A6:   i >= m;
      then a + b = 0 + b by A1,Def1
        .= 0 + 0 by A1,A6,Def3;
      hence thesis by A2,A4,Def4;
    end;
  end;
  hence thesis;
end;
