let m, k be Nat; :: thesis: ( k >= 2 implies for r being Tuple of m + 2,k -SD holds SDDec (Mmax r) >= SDDec r )
assume A1: k >= 2 ; :: thesis: for r being Tuple of m + 2,k -SD holds SDDec (Mmax r) >= SDDec r
let r be Tuple of m + 2,k -SD ; :: thesis: SDDec (Mmax r) >= SDDec r
for i being Nat st i in Seg (m + 2) holds
DigA ((Mmax r),i) >= DigA (r,i)
proof
let i be Nat; :: thesis: ( i in Seg (m + 2) implies DigA ((Mmax r),i) >= DigA (r,i) )
assume A2: i in Seg (m + 2) ; :: thesis: DigA ((Mmax r),i) >= DigA (r,i)
then A3: DigA ((Mmax r),i) = MmaxDigit (r,i) by Def4;
now :: thesis: DigA ((Mmax r),i) >= DigA (r,i)
per cases ( i >= m or i < m ) ;
suppose i >= m ; :: thesis: DigA ((Mmax r),i) >= DigA (r,i)
then DigA ((Mmax r),i) = r . i by A1, A2, A3, Def3
.= DigA (r,i) by A2, RADIX_1:def 3 ;
hence DigA ((Mmax r),i) >= DigA (r,i) ; :: thesis: verum
end;
suppose A4: i < m ; :: thesis: DigA ((Mmax r),i) >= DigA (r,i)
A5: DigA (r,i) = r . i by A2, RADIX_1:def 3;
A6: r . i is Element of k -SD by A2, RADIX_1:15;
DigA ((Mmax r),i) = (Radix k) - 1 by A1, A2, A3, A4, Def3;
hence DigA ((Mmax r),i) >= DigA (r,i) by A5, A6, RADIX_1:13; :: thesis: verum
end;
end;
end;
hence DigA ((Mmax r),i) >= DigA (r,i) ; :: thesis: verum
end;
hence SDDec (Mmax r) >= SDDec r by NAT_1:12, RADIX_5:13; :: thesis: verum