
theorem Th12:
  for m,k be Nat, r be Tuple of (m+2),k-SD st m >= 1 & k >= 2
holds SDDec(M0(r)) + SDDec(DecSD(0,m+2,k)) = SDDec(Mmin(r)) + SDDec(SDMax(m+2,m
  ,k))
proof
  let m,k be Nat, r be Tuple of (m+2),k-SD;
  assume that
A1: m >= 1 and
A2: k >= 2;
A3: m+2 > 1 by A1,Lm1;
  SDDec(M0(r)) + SDDec(SDMin(m+2,m,k)) = SDDec(Mmin(r)) + SDDec(DecSD(0,m+
  2,k)) by A2,Th11
    .= SDDec(Mmin(r)) + 0 by A3,RADIX_5:6;
  then
  SDDec(Mmin(r)) + SDDec(SDMax(m+2,m,k)) = SDDec(M0(r)) + (SDDec(SDMax(m+2
  ,m,k)) + SDDec(SDMin(m+2,m,k)))
    .= SDDec(M0(r)) + SDDec(DecSD(0,m+2,k)) by A2,A3,RADIX_5:17;
  hence thesis;
end;
