
theorem
  for n be Nat st n >= 1 holds for m,k be Nat st m in Seg n & k >= 2
  holds SDDec(Fmin(n,m,k)) = SDDec(SDMax(n,m,k)) + SDDec(DecSD(1,n,k))
proof
  let n be Nat;
A1: 1 in Seg 1 by FINSEQ_1:1;
  assume
A2: n >= 1;
  then
A3: 1 in Seg n by FINSEQ_1:1;
  let m,k be Nat;
  assume that
A4: m in Seg n and
A5: k >= 2;
A6: n >= m by A4,FINSEQ_1:1;
A7: m >= 1 by A4,FINSEQ_1:1;
  now
    per cases by A2,XXREAL_0:1;
    suppose
A8:   n = 1;
      then
A9:   m = 1 by A6,A7,XXREAL_0:1;
A10:  SDDec(Fmin(1,m,k)) = DigA(Fmin(1,m,k),1) by Th4
        .= FminDigit(m,k,1) by A1,Def6
        .= 1 by A5,A9,Def5;
A11:  DigA(SDMax(1,m,k),1) = SDMaxDigit(m,k,1) by A1,Def4
        .= 0 by A5,A6,A8,Def3;
      SDDec(SDMax(1,m,k) '+' DecSD(1,1,k)) = DigA(SDMax(1,m,k) '+' DecSD(
      1,1,k),1) by Th4
        .= Add(SDMax(1,m,k),DecSD(1,1,k),1,k) by A1,RADIX_1:def 14
        .= SD_Add_Data(DigA(SDMax(1,m,k),1)+DigA(DecSD(1,1,k),1),k) +
SD_Add_Carry(DigA(SDMax(1,m,k),1-'1)+DigA(DecSD(1,1,k),1-'1)) by A5,A1,
RADIX_1:def 13
        .= SD_Add_Data(DigA(SDMax(1,m,k),1)+DigA(DecSD(1,1,k),1),k) +
      SD_Add_Carry(DigA(SDMax(1,m,k),0)+DigA(DecSD(1,1,k),1-'1)) by XREAL_1:232
        .= SD_Add_Data(DigA(SDMax(1,m,k),1)+DigA(DecSD(1,1,k),1),k) +
      SD_Add_Carry(DigA(SDMax(1,m,k),0)+DigA(DecSD(1,1,k),0)) by XREAL_1:232
        .= SD_Add_Data(DigA(SDMax(1,m,k),1)+DigA(DecSD(1,1,k),1),k) +
      SD_Add_Carry(DigA(SDMax(1,m,k),0)+0) by RADIX_1:def 3
        .= SD_Add_Data(DigA(SDMax(1,m,k),1)+DigA(DecSD(1,1,k),1),k) +
      SD_Add_Carry(0+0) by RADIX_1:def 3
        .= SD_Add_Data(DigA(SDMax(1,m,k),1)+DigA(DecSD(1,1,k),1),k) + 0 by
RADIX_1:def 10
        .= SD_Add_Data(DigA(SDMax(1,m,k),1)+1,k) by A5,A1,Th7;
      then
A12:  SDDec(SDMax(1,m,k) '+' DecSD(1,1,k)) = 1 - SD_Add_Carry(1) * Radix
      (k) by A11,RADIX_1:def 11
        .= 1 - 0 * Radix(k) by RADIX_1:def 10
        .= 1;
      SDDec(SDMax(1,m,k)) + SDDec(DecSD(1,1,k)) = SDDec(SDMax(1,m,k) '+'
DecSD(1,1,k)) + SD_Add_Carry(DigA(SDMax(1,m,k),1)+DigA(DecSD(1,1,k),1)) *(Radix
      (k) |^ 1) by A5,RADIX_2:11
        .= 1 + SD_Add_Carry(0+1) *(Radix(k) |^ 1) by A5,A1,A11,A12,Th7
        .= 1 + 0 * (Radix(k) |^ 1) by RADIX_1:def 10;
      hence thesis by A8,A10;
    end;
    suppose
A13:  n > 1;
A14:  n in Seg n by A2,FINSEQ_1:1;
      then
A15:  DigA(SDMax(n,m,k),n) = SDMaxDigit(m,k,n) by Def4
        .= 0 by A5,A6,Def3;
A16:  for i be Nat st i in Seg n holds DigA(Fmin(n,m,k),i) = DigA(SDMax(n
      ,m,k) '+' DecSD(1,n,k),i)
      proof
        let i be Nat;
        assume
A17:    i in Seg n;
        then
A18:    DigA(SDMax(n,m,k) '+' DecSD(1,n,k),i) = Add(SDMax(n,m,k),DecSD(1,
        n,k),i,k) by RADIX_1:def 14
          .= SD_Add_Data(DigA(SDMax(n,m,k),i)+DigA(DecSD(1,n,k),i),k) +
SD_Add_Carry(DigA(SDMax(n,m,k),i-'1)+DigA(DecSD(1,n,k),i-'1)) by A5,A17,
RADIX_1:def 13;
A19:    DigA(SDMax(n,m,k),i) = SDMaxDigit(m,k,i) by A17,Def4;
A20:    DigA(Fmin(n,m,k),i) = FminDigit(m,k,i) by A17,Def6;
A21:    i >= 1 by A17,FINSEQ_1:1;
        now
          per cases;
          suppose
A22:        i >= m+1;
            then
A23:        i > m by NAT_1:13;
            then
A24:        DigA(SDMax(n,m,k),i) = 0 by A5,A19,Def3;
A25:        i > 1 by A7,A23,XXREAL_0:2;
            then
A26:        i -' 1 in Seg n & DigA(DecSD(1,n,k),i) = 0 by A5,A17,Th2,Th8;
            now
              per cases by A22,XXREAL_0:1;
              suppose
                i = m+1;
                then DigA(SDMax(n,m,k),i-'1) = DigA(SDMax(n,m,k),m) by NAT_D:34
                  .= SDMaxDigit(m,k,m) by A4,Def4
                  .= 0 by A5,Def3;
                then
                DigA(SDMax(n,m,k) '+' DecSD(1,n,k),i) = SD_Add_Data(0+0,k
                ) + 0 by A5,A18,A24,A26,Lm2
                  .= 0 by RADIX_1:19;
                hence thesis by A5,A20,A23,Def5;
              end;
              suppose
                i > m+1;
                then i - 1 > m + 1 - 1 by XREAL_1:14;
                then
A27:            i -' 1 > m by XREAL_0:def 2;
                i -' 1 in Seg n by A17,A25,Th2;
                then DigA(SDMax(n,m,k),i-'1) = SDMaxDigit(m,k,i-'1) by Def4
                  .= 0 by A5,A27,Def3;
                then
                DigA(SDMax(n,m,k) '+' DecSD(1,n,k),i) = SD_Add_Data(0+0,k
                ) + 0 by A5,A18,A24,A26,Lm2
                  .= 0 by RADIX_1:19;
                hence thesis by A5,A20,A23,Def5;
              end;
            end;
            hence thesis;
          end;
          suppose
            i < m+1;
            then
A28:        i <= m by NAT_1:13;
A29:        i >= 1 by A17,FINSEQ_1:1;
            now
              per cases by A29,XXREAL_0:1;
              suppose
A30:            i > 1;
                then
A31:            m > 1 by A28,XXREAL_0:2;
                then
A32:            m -' 1 < m by JORDAN5B:1;
A33:            m -' 1 in Seg n by A4,A31,Th2;
                then
A34:            m -' 1 >= 1 by FINSEQ_1:1;
                now
                  per cases by A28,XXREAL_0:1;
                  suppose
A35:                i = m;
                    then
A36:                DigA(SDMax(n,m,k),i) = 0 & DigA(DecSD(1,n,k),i) = 0
                    by A4,A5,A19,A30,Def3,Th8;
A37:                DigA(Fmin(n,m,k),i) = FminDigit(m, k,m) by A4,A35,Def6
                      .= 1 by A5,Def5;
A38:                DigA(SDMax(n,m,k),i-'1) = SDMaxDigit(m,k,m-'1) by A33,A35
,Def4
                      .= Radix(k) - 1 by A5,A34,A32,Def3;
A39:                i >= 1 + 1 by A30,INT_1:7;
                    now
                      per cases by A39,XXREAL_0:1;
                      suppose
                        i = 2;
                        then i -' 1 = 2 - 1 by XREAL_1:233;
                        then DigA(SDMax(n,m,k) '+' DecSD(1,n,k),i) =
SD_Add_Data(0+0,k) +SD_Add_Carry(Radix(k) - 1 + 1) by A5,A3,A18,A36,A38,Th7
                          .= 0 + SD_Add_Carry(Radix(k) - 1 + 1) by RADIX_1:19
                          .= 1 by A5,Th10;
                        hence thesis by A37;
                      end;
                      suppose
A40:                    i > 2;
                        then i - 1 > 2 - 1 by XREAL_1:14;
                        then
A41:                    i -' 1 > 1 by XREAL_0:def 2;
                        i > 1 by A40,XXREAL_0:2;
                        then i -' 1 in Seg n by A17,Th2;
                        then DigA(SDMax(n,m,k) '+' DecSD(1,n,k),i) =
SD_Add_Data(0+0,k) + SD_Add_Carry(Radix(k) - 1 + 0) by A5,A18,A36,A38,A41,Th8
                          .= 0 + SD_Add_Carry(Radix(k) - 1) by RADIX_1:19
                          .= 1 by A5,Lm1;
                        hence thesis by A37;
                      end;
                    end;
                    hence thesis;
                  end;
                  suppose
A42:                i < m;
                    i -' 1 < i by A30,JORDAN5B:1;
                    then
A43:                i -' 1 < m by A42,XXREAL_0:2;
A44:                DigA(DecSD(1,n,k),i) = 0 by A5,A17,A30,Th8;
A45:                i -' 1 in Seg n by A17,A30,Th2;
A46:                i >= 1 + 1 by A30,INT_1:7;
                    now
                      per cases by A46,XXREAL_0:1;
                      suppose
                        i = 2;
                        then
A47:                    i -' 1 = 2 - 1 by XREAL_1:233;
                        then
A48:                    DigA(DecSD(1,n,k),i-'1) = 1 by A5,A3,Th7;
                        DigA(SDMax(n,m,k),i-'1) = SDMaxDigit(m,k,i-'1) by A45
,Def4
                          .= Radix(k) - 1 by A5,A43,A47,Def3;
                        then DigA(SDMax(n,m,k) '+' DecSD(1,n,k),i) =
SD_Add_Data(Radix(k)-1,k) +SD_Add_Carry(Radix(k) + 1 - 1) by A5,A18,A19,A21,A42
,A44,A48,Def3
                          .= SD_Add_Data(Radix(k)-1,k) + 1 by A5,Th10
                          .= (Radix(k)-1) - SD_Add_Carry(Radix(k)-1)*Radix(k
                        ) + 1 by RADIX_1:def 11
                          .= (Radix(k)-1) - 1*Radix(k) + 1 by A5,Lm1
                          .= 0;
                        hence thesis by A5,A20,A42,Def5;
                      end;
                      suppose
A49:                    i > 2;
                        then i - 1 > 2 - 1 by XREAL_1:14;
                        then
A50:                    i -' 1 > 1 by XREAL_0:def 2;
                        i > 1 by A49,XXREAL_0:2;
                        then i -' 1 in Seg n by A17,Th2;
                        then
A51:                    DigA(DecSD(1,n,k),i-'1) = 0 by A5,A50,Th8;
                        DigA(SDMax(n,m,k),i-'1) = SDMaxDigit(m,k,i-'1) by A45
,Def4
                          .= Radix(k) - 1 by A5,A43,A50,Def3;
                        then DigA(SDMax(n,m,k) '+' DecSD(1,n,k),i) =
SD_Add_Data(Radix(k)-1+0,k) +SD_Add_Carry(Radix(k) - 1 + 0) by A5,A18,A19,A21
,A42,A44,A51,Def3
                          .= SD_Add_Data(Radix(k)-1,k) + 1 by A5,Lm1
                          .= (Radix(k)-1) - SD_Add_Carry(Radix(k)-1)*Radix(k
                        ) + 1 by RADIX_1:def 11
                          .= (Radix(k)-1) - 1*Radix(k) + 1 by A5,Lm1
                          .= 0;
                        hence thesis by A5,A20,A42,Def5;
                      end;
                    end;
                    hence thesis;
                  end;
                end;
                hence thesis;
              end;
              suppose
A52:            i = 1;
                then
A53:            DigA(SDMax(n,m,k),i-'1) = 0 & DigA(DecSD(1,n,k),i-'1) = 0
                by NAT_2:8,RADIX_1:def 3;
                now
                  per cases by A28,XXREAL_0:1;
                  suppose
A54:                i < m;
                    then DigA(SDMax(n,m,k),i)=Radix(k)-1 by A5,A19,A52,Def3;
                    then DigA(SDMax(n,m,k) '+' DecSD(1,n,k),i) = SD_Add_Data(
Radix(k)-1+1,k) + SD_Add_Carry(0+0) by A5,A3,A18,A52,A53,Th7
                      .= Radix(k) - SD_Add_Carry(Radix(k)) * Radix(k) by
RADIX_1:18,def 11
                      .= Radix(k) - 1 * Radix(k) by A5,Th10
                      .= 0;
                    hence thesis by A5,A20,A54,Def5;
                  end;
                  suppose
A55:                i = m;
                    then DigA(SDMax(n,m,k),i) = 0 by A5,A19,Def3;
                    then
                    DigA(SDMax(n,m,k) '+' DecSD(1,n,k),i) = SD_Add_Data(1
                    +0,k)+SD_Add_Carry(0+0) by A5,A3,A18,A52,A53,Th7
                      .= 1 - SD_Add_Carry(1) * Radix(k) by RADIX_1:18,def 11
                      .= 1 - 0 * Radix(k) by RADIX_1:def 10
                      .= 1;
                    hence thesis by A5,A20,A55,Def5;
                  end;
                end;
                hence thesis;
              end;
            end;
            hence thesis;
          end;
        end;
        hence thesis;
      end;
      SDDec(SDMax(n,m,k)) + SDDec(DecSD(1,n,k)) = SDDec(SDMax(n,m,k) '+'
DecSD(1,n,k)) + SD_Add_Carry(DigA(SDMax(n,m,k),n)+DigA(DecSD(1,n,k),n)) * (
      Radix(k) |^ n) by A2,A5,RADIX_2:11
        .= SDDec(SDMax(n,m,k) '+' DecSD(1,n,k)) + 0 * (Radix(k) |^ n) by A5,A13
,A14,A15,Th8,RADIX_1:18;
      hence thesis by A2,A16,Th12;
    end;
  end;
  hence thesis;
end;
