
theorem Th13:
  for m,k be Nat st m >= 1 & k >= 2 holds for r be Tuple of (m+2),
  k-SD holds SDDec(M0(r)) < SDDec(Mmin(r)) + SDDec(Fmin(m+2,m,k))
proof
  let m,k be Nat;
  assume that
A1: m >= 1 and
A2: k >= 2;
  let r be Tuple of (m+2),k-SD;
A3: m+2 > 1 by A1,Lm1;
A4: SDDec(Mmin(r)) + SDDec(SDMax(m+2,m,k)) = SDDec(M0(r)) + SDDec(DecSD(0,m+
  2,k)) by A1,A2,Th12
    .= SDDec(M0(r)) + 0 by A3,RADIX_5:6;
A5: SDDec(M0(r)) + 1 > SDDec(M0(r)) + 0 by XREAL_1:8;
  m in Seg (m+2) by A1,FINSEQ_3:9;
  then
  SDDec(Fmin(m+2,m,k)) = SDDec(SDMax(m+2,m,k)) + SDDec(DecSD(1,m+2,k)) by A2,A3
,RADIX_5:18
    .= SDDec(SDMax(m+2,m,k)) + 1 by A2,A3,RADIX_5:9;
  hence thesis by A4,A5;
end;
