reserve n,m,k for Nat,
        p,q for n-element XFinSequence of NAT,
        i1,i2,i3,i4,i5,i6 for Element of n,
        a,b,c,d,e for Integer;

theorem
  for a,b,i1,i2 holds
    {p: p.i1 = a*p.i2+b} is diophantine Subset of n -xtuples_of NAT
proof
  let a,b be Integer,i1,i2;
  defpred P[XFinSequence of NAT] means $1.i1 = a * $1.i2 +b;
  defpred Q[XFinSequence of NAT] means 1 * ($1.i1) = a * $1.i2+b;
  A1:for p holds P[p] iff Q[p];
  {p: P[p]} = {q:Q[q]} from Eq(A1);
  hence thesis by Th6;
end;
