let m, k be Nat; :: thesis: for r being Tuple of (m + 2),(k -SD ) st m >= 1 & k >= 2 holds
(SDDec (M0 r)) + (SDDec (Mmask r)) = (SDDec r) + (SDDec (DecSD 0 ,(m + 2),k))
let r be Tuple of (m + 2),(k -SD ); :: thesis: ( m >= 1 & k >= 2 implies (SDDec (M0 r)) + (SDDec (Mmask r)) = (SDDec r) + (SDDec (DecSD 0 ,(m + 2),k)) )
assume A1: ( m >= 1 & k >= 2 ) ; :: thesis: (SDDec (M0 r)) + (SDDec (Mmask r)) = (SDDec r) + (SDDec (DecSD 0 ,(m + 2),k))
A2: m + 2 >= 1 by NAT_1:12;
for i being Nat holds
( not i in Seg (m + 2) or ( DigA (M0 r),i = DigA r,i & DigA (Mmask r),i = 0 ) or ( DigA (Mmask r),i = DigA r,i & DigA (M0 r),i = 0 ) )

proof

let i be Nat; :: thesis: ( not i in Seg (m + 2) or ( DigA (M0 r),i = DigA r,i & DigA (Mmask r),i = 0 ) or ( DigA (Mmask r),i = DigA r,i & DigA (M0 r),i = 0 ) )
assume A3: i in Seg (m + 2) ; :: thesis: ( ( DigA (M0 r),i = DigA r,i & DigA (Mmask r),i = 0 ) or ( DigA (Mmask r),i = DigA r,i & DigA (M0 r),i = 0 ) )

now

per cases ( i < m or i >= m ) ;

suppose A4: i < m ; :: thesis: ( ( DigA (M0 r),i = DigA r,i & DigA (Mmask r),i = 0 ) or ( DigA (Mmask r),i = DigA r,i & DigA (M0 r),i = 0 ) )

A5: DigA (Mmask r),i = MmaskDigit r,i by A3, Def9
.= r . i by A3, A4, Def8
.= DigA r,i by A3, RADIX_1:def 3 ;
DigA (M0 r),i = M0Digit r,i by A3, Def2
.= 0 by A3, A4, Def1 ;
hence ( ( DigA (M0 r),i = DigA r,i & DigA (Mmask r),i = 0 ) or ( DigA (Mmask r),i = DigA r,i & DigA (M0 r),i = 0 ) ) by A5; :: thesis: verum

end;

suppose A6: i >= m ; :: thesis: ( ( DigA (M0 r),i = DigA r,i & DigA (Mmask r),i = 0 ) or ( DigA (Mmask r),i = DigA r,i & DigA (M0 r),i = 0 ) )

A7: DigA (Mmask r),i = MmaskDigit r,i by A3, Def9
.= 0 by A3, A6, Def8 ;
DigA (M0 r),i = M0Digit r,i by A3, Def2
.= r . i by A3, A6, Def1
.= DigA r,i by A3, RADIX_1:def 3 ;
hence ( ( DigA (M0 r),i = DigA r,i & DigA (Mmask r),i = 0 ) or ( DigA (Mmask r),i = DigA r,i & DigA (M0 r),i = 0 ) ) by A7; :: thesis: verum

end;

end;

end;

hence ( ( DigA (M0 r),i = DigA r,i & DigA (Mmask r),i = 0 ) or ( DigA (Mmask r),i = DigA r,i & DigA (M0 r),i = 0 ) ) ; :: thesis: verum

end;

hence (SDDec (M0 r)) + (SDDec (Mmask r)) = (SDDec r) + (SDDec (DecSD 0 ,(m + 2),k)) by A1, A2, RADIX_5:15; :: thesis: verum