
theorem
  for n,m,k be Nat st m in Seg n & k >= 2 holds SDDec(Fmin(n+1,m+1,k)) =
  SDDec(Fmin(n+1,m,k)) + SDDec(Fmax(n+1,m,k))
proof
  let n,m,k be Nat;
  assume that
A1: m in Seg n and
A2: k >= 2;
  n >= m by A1,FINSEQ_1:1;
  then
A3: n + 1 > m by NAT_1:13;
A4: n+1 in Seg(n+1) by FINSEQ_1:3;
  then
A5: DigA(Fmin(n+1,m,k),n+1) = FminDigit(m,k,n+1) by Def6
    .= 0 by A2,A3,Def5;
A6: m in Seg (n+1) by A1,FINSEQ_2:8;
A7: m >= 1 by A1,FINSEQ_1:1;
A8: for i be Nat st i in Seg (n+1) holds DigA(Fmin(n+1,m+1,k),i) = DigA(
  Fmin(n+1,m,k) '+' Fmax(n+1,m,k),i)
  proof
    let i be Nat;
A9: m + 1 > m by NAT_1:13;
    assume
A10: i in Seg (n+1);
    then
A11: DigA(Fmin(n+1,m,k) '+' Fmax(n+1,m,k),i) = Add(Fmin(n+1,m,k),Fmax(n+1,
    m,k),i,k) by RADIX_1:def 14
      .= SD_Add_Data(DigA(Fmin(n+1,m,k),i)+DigA(Fmax(n+1,m,k),i),k) +
    SD_Add_Carry(DigA(Fmin(n+1,m,k),i-'1)+DigA(Fmax(n+1,m,k),i-'1)) by A2,A10,
RADIX_1:def 13;
A12: DigA(Fmin(n+1,m+1,k),i) = FminDigit(m+1,k,i) by A10,Def6;
A13: DigA(Fmin(n+1,m,k),i) = FminDigit(m,k,i) by A10,Def6;
A14: DigA(Fmax(n+1,m,k),i) = FmaxDigit(m,k,i) by A10,Def8;
A15: i >= 1 by A10,FINSEQ_1:1;
    now
      per cases;
      suppose
A16:    i >= m+1;
        now
          per cases;
          suppose
A17:        i = m+1;
            then
A18:        DigA(Fmin(n+1,m+1,k),i) = FminDigit(m+1,k,m+1) by A10,Def6
              .= 1 by A2,Def5;
A19:        DigA(Fmax(n+1,m,k),i-'1) = DigA(Fmax(n+1,m,k),m) by A17,NAT_D:34
              .= FmaxDigit(m,k,m) by A6,Def8
              .= Radix(k) - 1 by A2,Def7;
A20:        DigA(Fmax(n+1,m,k),i) = FmaxDigit(m,k,m+1) by A10,A17,Def8
              .= 0 by A2,A9,Def7;
A21:        DigA(Fmin(n+1,m,k),i-'1) = DigA(Fmin(n+1,m,k),m) by A17,NAT_D:34
              .= FminDigit(m,k,m) by A6,Def6
              .= 1 by A2,Def5;
            DigA(Fmin(n+1,m,k),i) = FminDigit(m,k,m+1) by A10,A17,Def6
              .= 0 by A2,A9,Def5;
            then
            DigA(Fmin(n+1,m,k) '+' Fmax(n+1,m,k),i) = 0 + SD_Add_Carry(1+
            (Radix(k)-1)) by A11,A20,A21,A19,RADIX_1:19
              .= 1 by A2,Th10;
            hence thesis by A18;
          end;
          suppose
A22:        i <> m + 1;
            then i > m + 1 by A16,XXREAL_0:1;
            then i - 1 > m + 1 - 1 by XREAL_1:14;
            then
A23:        i -' 1 > m by XREAL_0:def 2;
            i > m by A16,NAT_1:13;
            then i > 1 by A7,XXREAL_0:2;
            then
A24:        i -' 1 in Seg (n+1) by A10,Th2;
            then
A25:        DigA(Fmin(n+1,m,k),i-'1) = FminDigit(m,k,i-'1) by Def6
              .= 0 by A2,A23,Def5;
A26:        DigA(Fmin(n+1,m+1,k),i) = 0 by A2,A12,A22,Def5;
A27:        DigA(Fmax(n+1,m,k),i-'1) = FmaxDigit(m,k,i-'1) by A24,Def8
              .= 0 by A2,A23,Def7;
            DigA(Fmin(n+1,m,k),i) = 0 & DigA(Fmax(n+1,m,k),i) = 0 by A2,A9,A13
,A14,A16,Def5,Def7;
            hence thesis by A11,A25,A27,A26,RADIX_1:18,19;
          end;
        end;
        hence thesis;
      end;
      suppose
        i < m+1;
        then
A28:    i <= m by NAT_1:13;
        now
          per cases by A15,XXREAL_0:1;
          suppose
A29:        i > 1;
            then
A30:        m > 1 by A28,XXREAL_0:2;
            then
A31:        m -' 1 < m by JORDAN5B:1;
A32:        m -' 1 in Seg (n+1) by A6,A30,Th2;
            now
              per cases by A28,XXREAL_0:1;
              suppose
A33:            i = m;
                then
A34:            DigA(Fmax(n+1,m,k),i-'1) = FmaxDigit(m,k,m-'1) by A32,Def8
                  .= 0 by A2,A31,Def7;
A35:            DigA(Fmax(n+1,m,k),i) = Radix(k) - 1 by A2,A14,A33,Def7;
A36:            DigA(Fmin(n+1,m+1,k),i) = FminDigit(m+1,k,m) by A10,A33,Def6
                  .= 0 by A2,A9,Def5;
                DigA(Fmin(n+1,m,k),i-'1) = FminDigit(m,k,m-'1) by A32,A33,Def6
                  .= 0 by A2,A31,Def5;
                then
                DigA(Fmin(n+1,m,k) '+' Fmax(n+1,m,k),i) = SD_Add_Data(1 +
                (Radix(k) - 1),k) + 0 by A2,A11,A13,A33,A35,A34,Def5,RADIX_1:18
                  .= 0 by A2,Th11;
                hence thesis by A36;
              end;
              suppose
A37:            i < m;
A38:            i -' 1 >= 1 by A29,NAT_D:49;
A39:            i -' 1 < i by A15,JORDAN5B:1;
                then i -' 1 < m by A37,XXREAL_0:2;
                then i -' 1 <= n+1 by A3,XXREAL_0:2;
                then
A40:            i -' 1 in Seg (n+1) by A38,FINSEQ_1:1;
                then
A41:            DigA(Fmax(n+1,m,k),i-'1) = FmaxDigit(m,k,i-'1) by Def8
                  .= 0 by A2,A37,A39,Def7;
A42:            DigA(Fmax(n+1,m,k),i) = 0 by A2,A14,A37,Def7;
A43:            i < m + 1 by A37,NAT_1:13;
                DigA(Fmin(n+1,m,k),i-'1) = FminDigit(m,k,i-'1) by A40,Def6
                  .= 0 by A2,A37,A39,Def5;
                then
                DigA(Fmin(n+1,m,k) '+' Fmax(n+1,m,k),i) = SD_Add_Data(0+0
                ,k) + SD_Add_Carry(0+0) by A2,A11,A13,A37,A42,A41,Def5
                  .= 0 + 0 by RADIX_1:18,19;
                hence thesis by A2,A12,A43,Def5;
              end;
            end;
            hence thesis;
          end;
          suppose
A44:        i = 1;
            then
A45:        DigA(Fmax(n+1,m,k),i-'1) = 0 by NAT_2:8,RADIX_1:def 3;
            now
              per cases by A28,XXREAL_0:1;
              suppose
A46:            i < m;
                then DigA(Fmin(n+1,m,k),i) = 0 & DigA(Fmax(n+1,m,k),i) = 0 by
A2,A13,A14,Def5,Def7;
                then
A47:            DigA(Fmin(n+1,m,k) '+' Fmax(n+1,m,k),i) = SD_Add_Data(0+0
                ,k) + SD_Add_Carry(0+0) by A11,A44,A45,NAT_2:8,RADIX_1:def 3
                  .= 0 + 0 by RADIX_1:18,19;
                i < m + 1 by A46,NAT_1:13;
                hence thesis by A2,A12,A47,Def5;
              end;
              suppose
A48:            i = m;
                then
                DigA(Fmin(n+1,m,k),i) = 1 & DigA(Fmax(n+1,m,k),i) = Radix
                (k) - 1 by A2,A13,A14,Def5,Def7;
                then
A49:            DigA(Fmin(n+1,m,k) '+' Fmax(n+1,m,k),i) = SD_Add_Data(1 +
                (Radix(k) - 1),k) + 0 by A11,A44,A45,NAT_2:8,RADIX_1:18,def 3
                  .= 0 by A2,Th11;
                i < m + 1 by A48,NAT_1:13;
                hence thesis by A2,A12,A49,Def5;
              end;
            end;
            hence thesis;
          end;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
A50: DigA(Fmax(n+1,m,k),n+1) = FmaxDigit(m,k,n+1) by A4,Def8
    .= 0 by A2,A3,Def7;
  SDDec(Fmin(n+1,m,k)) + SDDec(Fmax(n+1,m,k)) = SDDec(Fmin(n+1,m,k) '+'
Fmax(n+1,m,k)) + SD_Add_Carry(DigA(Fmin(n+1,m,k),n+1)+DigA(Fmax(n+1,m,k),n+1))
  * (Radix(k) |^ (n+1)) by A2,NAT_1:11,RADIX_2:11
    .= SDDec(Fmin(n+1,m,k) '+' Fmax(n+1,m,k)) + 0 * (Radix(k) |^ (n+1)) by A5
,A50,RADIX_1:18;
  hence thesis by A8,Th12,NAT_1:11;
end;
