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 Th25:
  for a,b being Nat,i1,i2,i3 holds
    {p: p.i1 = (a*p.i2+b) * p.i3} is diophantine Subset of n -xtuples_of NAT
proof
  let a,b be Nat;
  deffunc F1(Nat,Nat,Nat) = a*$1+ b;
A1: for n,i1,i2,i3,i4 holds {p : F1(p.i1,p.i2,p.i3) = p.i4}
       is diophantine Subset of n -xtuples_of NAT by HILB10_3:15;
  defpred P1[Nat,Nat,natural object,Nat,Nat,Nat] means 1*$1 = 1 * $3 *$2;
A2: for n,i1,i2,i3,i4,i5,i6 holds {p : P1[p.i1,p.i2,p.i3,p.i4,p.i5,p.i6]}
    is diophantine Subset of n -xtuples_of NAT by HILB10_3:9;
A3: for n,i1,i2,i3,i4,i5 holds
    {p: P1[p.i1,p.i2,F1(p.i3,p.i4,p.i5),p.i3,p.i4,p.i5]}
    is diophantine Subset of n -xtuples_of NAT from HILB10_3:sch 4(A2,A1);
  let i1,i2,i3;
  defpred Q1[XFinSequence of NAT] means
    P1[$1.i1,$1.i3,a*$1.i2+b,$1.i3,$1.i3,$1.i3];
  defpred Q2[XFinSequence of NAT] means $1.i1 = (a*$1.i2+b) * $1.i3;
A4: for p holds Q1[p] iff Q2[p];
  {p : Q1[p]} = {q :Q2[q]} from HILB10_3:sch 2(A4);
  hence thesis by A3;
end;
