let m, k be Nat; :: thesis: ( k >= 2 implies for r being Tuple of ,k -SD holds SDDec (Mmax r) >= SDDec r )
assume A1: k >= 2 ; :: thesis: for r being Tuple of ,k -SD holds SDDec (Mmax r) >= SDDec r
let r be Tuple of ,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

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_1:18;
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_1:15; :: 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