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 Th8:
  for a,b,c,i1,i2 holds
    {p: a*p.i1 >= b*p.i2+c} is diophantine Subset of n -xtuples_of NAT
proof
  let a,b,c be Integer,i1,i2 be Element of n;
  defpred P[XFinSequence of NAT] means a* $1.i1 > b*$1.i2+c;
  defpred Q[XFinSequence of NAT] means a* $1.i1 = b*$1.i2+c;
  defpred R[XFinSequence of NAT] means P[$1] or Q[$1];
  defpred S[XFinSequence of NAT] means a* $1.i1 >= b*$1.i2+c;
  A1:{p: P[p]} is diophantine Subset of n -xtuples_of NAT by Th7;
  A2:{p: Q[p]} is diophantine Subset of n -xtuples_of NAT by Th6;
  A3: {p: P[p] or Q[p]} is diophantine Subset of n -xtuples_of NAT
  from UnionDiophantine(A1,A2);
  A4:R[p] iff S[p] by XXREAL_0:1;
  {p: R[p]} = {q: S[q]} from Eq(A4);
  hence thesis by A3;
end;
