
theorem
  for n be Nat st n >= 1 holds for k be Nat, tx,ty be Tuple of n,k-SD st
(for i be Nat st i in Seg n holds DigA(tx,i) >= DigA(ty,i)) holds SDDec(tx) >=
  SDDec(ty)
proof
  defpred P[Nat] means for k be Nat, tx,ty be Tuple of $1,k-SD st (for i be
Nat st i in Seg $1 holds DigA(tx,i) >= DigA(ty,i)) holds SDDec(tx) >= SDDec(ty)
  ;
A1: for n be Nat st n >= 1 & P[n] holds P[n+1]
  proof
    let n be Nat;
    assume that
    n >= 1 and
A2: P[n];
    let k be Nat, tx,ty be Tuple of (n+1),k-SD;
    assume
A3: for i be Nat st i in Seg (n+1) holds DigA(tx,i) >= DigA(ty,i);
    reconsider n as Element of NAT by ORDINAL1:def 12;
    deffunc F(Nat) = DigB(tx,$1);
    consider txn being FinSequence of INT such that
A4: len txn = n and
A5: for i be Nat st i in dom txn holds txn.i = F(i) from FINSEQ_2:sch
    1;
    deffunc F(Nat) = DigB(ty,$1);
    consider tyn being FinSequence of INT such that
A6: len tyn = n and
A7: for i be Nat st i in dom tyn holds tyn.i = F(i) from FINSEQ_2:sch
    1;
A8: dom tyn = Seg n by A6,FINSEQ_1:def 3;
    rng tyn c= k-SD
    proof
      let z be object;
      assume z in rng tyn;
      then consider yy be object such that
A9:   yy in dom tyn and
A10:  z = tyn.yy by FUNCT_1:def 3;
      reconsider yy as Element of NAT by A9;
      dom tyn = Seg n by A6,FINSEQ_1:def 3;
      then yy in Seg (n+1) by A9,FINSEQ_2:8;
      then
A11:  DigA(ty,yy) is Element of k-SD by RADIX_1:16;
      z = DigB(ty,yy) by A7,A9,A10
        .= DigA(ty,yy) by RADIX_1:def 4;
      hence thesis by A11;
    end;
    then reconsider tyn as FinSequence of k-SD by FINSEQ_1:def 4;
A12: for i be Nat st i in Seg n holds tyn.i = ty.i
    proof
      let i be Nat;
      assume
A13:  i in Seg n;
      then
A14:  i in Seg (n+1) by FINSEQ_2:8;
      tyn.i = DigB(ty,i) by A7,A8,A13
        .= DigA(ty,i) by RADIX_1:def 4;
      hence thesis by A14,RADIX_1:def 3;
    end;
    tyn is Element of n-tuples_on (k-SD) by A6,FINSEQ_2:92;
    then reconsider tyn as Tuple of n,k-SD;
A15: dom txn = Seg n by A4,FINSEQ_1:def 3;
    rng txn c= k-SD
    proof
      let z be object;
      assume z in rng txn;
      then consider xx be object such that
A16:  xx in dom txn and
A17:  z = txn.xx by FUNCT_1:def 3;
      reconsider xx as Element of NAT by A16;
      dom txn = Seg n by A4,FINSEQ_1:def 3;
      then xx in Seg (n+1) by A16,FINSEQ_2:8;
      then
A18:  DigA(tx,xx) is Element of k-SD by RADIX_1:16;
      z = DigB(tx,xx) by A5,A16,A17
        .= DigA(tx,xx) by RADIX_1:def 4;
      hence thesis by A18;
    end;
    then reconsider txn as FinSequence of k-SD by FINSEQ_1:def 4;
A19: for i be Nat st i in Seg n holds txn.i = tx.i
    proof
      let i be Nat;
      assume
A20:  i in Seg n;
      then
A21:  i in Seg (n+1) by FINSEQ_2:8;
      txn.i = DigB(tx,i) by A5,A15,A20
        .= DigA(tx,i) by RADIX_1:def 4;
      hence thesis by A21,RADIX_1:def 3;
    end;
    txn is Element of n-tuples_on (k-SD) by A4,FINSEQ_2:92;
    then reconsider txn as Tuple of n,k-SD;
    for i be Nat st i in Seg n holds DigA(txn,i) >= DigA(tyn,i)
    proof
      let i be Nat;
      assume
A22:  i in Seg n;
      then
A23:  DigA(tyn,i) = tyn.i by RADIX_1:def 3
        .= DigB(ty,i) by A7,A8,A22
        .= DigA(ty,i) by RADIX_1:def 4;
      DigA(txn,i) = txn.i by A22,RADIX_1:def 3
        .= DigB(tx,i) by A5,A15,A22
        .= DigA(tx,i) by RADIX_1:def 4;
      hence thesis by A3,A22,A23,FINSEQ_2:8;
    end;
    then
A24: SDDec(txn) >= SDDec(tyn) by A2;
    (n+1) in Seg (n+1) by FINSEQ_1:4;
    then
A25: (Radix(k) |^ n)*DigA(tx,n+1) >= (Radix(k) |^ n)*DigA(ty,n+1) by A3,
XREAL_1:64;
A26: SDDec(tyn) + (Radix(k) |^ n)*DigA(ty,n+1) = SDDec(ty) by A12,RADIX_2:10;
    SDDec(txn) + (Radix(k) |^ n)*DigA(tx,n+1) = SDDec(tx) by A19,RADIX_2:10;
    hence thesis by A26,A24,A25,XREAL_1:7;
  end;
A27: P[1]
  proof
    let k be Nat, tx,ty be Tuple of 1,k-SD;
    assume
A28: for i be Nat st i in Seg 1 holds DigA(tx,i) >= DigA(ty,i);
A29: SDDec(ty) = DigA(ty,1) by Th4;
    1 in Seg 1 & SDDec(tx) = DigA(tx,1) by Th4,FINSEQ_1:1;
    hence thesis by A28,A29;
  end;
  for n be Nat st n >= 1 holds P[n] from NAT_1:sch 8(A27,A1);
  hence thesis;
end;
