reserve i,j,n,n1,n2,m,k,l,u for Nat,
        r,r1,r2 for Real,
        x,y for Integer,
        a,b for non trivial Nat,
        F for XFinSequence,
        cF,cF1,cF2 for complex-valued XFinSequence,
        c,c1,c2 for Complex;
reserve x,y,x1,u,w for Nat;
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,d,f for Integer;

theorem
  for a,i1,i2,i3 holds
    {p: p.i1 = a*p.i2*p.i3} is diophantine Subset of n -xtuples_of NAT
proof
  let a,i1,i2,i3;
  defpred Q1[XFinSequence of NAT] means 1*$1.i1 = a*$1.i2*$1.i3;
  defpred Q2[XFinSequence of NAT] means $1.i1 = a*$1.i2*$1.i3;
A1:for p holds Q1[p] iff Q2[p];
  {p:Q1[p]} = {q:Q2[q]} from HILB10_3:sch 2(A1);
  hence thesis by HILB10_3:9;
end;
