reserve n for non zero Nat,
  j,k,l,m for Nat,
  g,h,i for Integer;

theorem
  for x,y be Tuple of n,BOOLEAN st x = 0*n & y = 0*n holds x,y are_summable
proof
  let x,y be Tuple of n,BOOLEAN such that
A1: x = 0*n and
A2: y = 0*n;
A3: 1 <= n by NAT_1:14;
  len x = n by CARD_1:def 7;
  then x/.n = (0*n).n by A1,A3,FINSEQ_4:15
    .= FALSE;
  then add_ovfl(x,y) = FALSE '&' FALSE 'or' FALSE '&' (carry(x,y)/.n) 'or'
  FALSE '&' (carry(x,y)/.n) by A1,A2,BINARITH:def 6
    .= FALSE;
  hence thesis by BINARITH:def 7;
end;
