let m, k be Nat; :: thesis: ( k >= 2 implies for r being Tuple of m + 2,k -SD holds (SDDec (Mmin r)) + (SDDec (DecSD 0 ,(m + 2),k)) = (SDDec (M0 r)) + (SDDec (SDMin (m + 2),m,k)) )
assume A1: k >= 2 ; :: thesis: for r being Tuple of m + 2,k -SD holds (SDDec (Mmin r)) + (SDDec (DecSD 0 ,(m + 2),k)) = (SDDec (M0 r)) + (SDDec (SDMin (m + 2),m,k))
let r be Tuple of m + 2,k -SD ; :: thesis: (SDDec (Mmin r)) + (SDDec (DecSD 0 ,(m + 2),k)) = (SDDec (M0 r)) + (SDDec (SDMin (m + 2),m,k))
A2: for i being Nat holds
( not i in Seg (m + 2) or ( DigA (M0 r),i = DigA (Mmin r),i & DigA (SDMin (m + 2),m,k),i = 0 ) or ( DigA (SDMin (m + 2),m,k),i = DigA (Mmin 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 (Mmin r),i & DigA (SDMin (m + 2),m,k),i = 0 ) or ( DigA (SDMin (m + 2),m,k),i = DigA (Mmin r),i & DigA (M0 r),i = 0 ) )
assume A3: i in Seg (m + 2) ; :: thesis: ( ( DigA (M0 r),i = DigA (Mmin r),i & DigA (SDMin (m + 2),m,k),i = 0 ) or ( DigA (SDMin (m + 2),m,k),i = DigA (Mmin r),i & DigA (M0 r),i = 0 ) )
then A4: DigA (Mmin r),i = MminDigit r,i by Def6;
A5: i >= 1 by A3, FINSEQ_1:3;

now

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

suppose A6: i < m ; :: thesis: ( ( DigA (M0 r),i = DigA (Mmin r),i & DigA (SDMin (m + 2),m,k),i = 0 ) or ( DigA (SDMin (m + 2),m,k),i = DigA (Mmin r),i & DigA (M0 r),i = 0 ) )

A7: DigA (M0 r),i = M0Digit r,i by A3, Def2
.= 0 by A3, A6, Def1 ;
DigA (Mmin r),i = (- (Radix k)) + 1 by A1, A3, A4, A6, Def5
.= SDMinDigit m,k,i by A1, A5, A6, RADIX_5:def 1
.= DigA (SDMin (m + 2),m,k),i by A3, RADIX_5:def 2 ;
hence ( ( DigA (M0 r),i = DigA (Mmin r),i & DigA (SDMin (m + 2),m,k),i = 0 ) or ( DigA (SDMin (m + 2),m,k),i = DigA (Mmin r),i & DigA (M0 r),i = 0 ) ) by A7; :: thesis: verum

end;

suppose A8: i >= m ; :: thesis: ( ( DigA (M0 r),i = DigA (Mmin r),i & DigA (SDMin (m + 2),m,k),i = 0 ) or ( DigA (SDMin (m + 2),m,k),i = DigA (Mmin r),i & DigA (M0 r),i = 0 ) )

A9: DigA (SDMin (m + 2),m,k),i = SDMinDigit m,k,i by A3, RADIX_5:def 2
.= 0 by A1, A8, RADIX_5:def 1 ;
DigA (Mmin r),i = r . i by A1, A3, A4, A8, Def5
.= M0Digit r,i by A3, A8, Def1
.= DigA (M0 r),i by A3, Def2 ;
hence ( ( DigA (M0 r),i = DigA (Mmin r),i & DigA (SDMin (m + 2),m,k),i = 0 ) or ( DigA (SDMin (m + 2),m,k),i = DigA (Mmin r),i & DigA (M0 r),i = 0 ) ) by A9; :: thesis: verum

end;

end;

end;

hence ( ( DigA (M0 r),i = DigA (Mmin r),i & DigA (SDMin (m + 2),m,k),i = 0 ) or ( DigA (SDMin (m + 2),m,k),i = DigA (Mmin r),i & DigA (M0 r),i = 0 ) ) ; :: thesis: verum

end;

m + 2 >= 1 by NAT_1:12;
hence (SDDec (Mmin r)) + (SDDec (DecSD 0 ,(m + 2),k)) = (SDDec (M0 r)) + (SDDec (SDMin (m + 2),m,k)) by A1, A2, RADIX_5:15; :: thesis: verum