let n be Nat; :: thesis: for a, b being Nat
for i1, i2, i3 being Element of n holds { p where p is n -element XFinSequence of NAT : p . i1 = ((a * (p . i2)) + b) * (p . i3) } is diophantine Subset of ()

let a, b be Nat; :: thesis: for i1, i2, i3 being Element of n holds { p where p is n -element XFinSequence of NAT : p . i1 = ((a * (p . i2)) + b) * (p . i3) } is diophantine Subset of ()
deffunc H1( Nat, Nat, Nat) -> set = (a * \$1) + b;
A1: for n being Nat
for i1, i2, i3, i4 being Element of n holds { p where p is b1 -element XFinSequence of NAT : H1(p . i1,p . i2,p . i3) = p . i4 } is diophantine Subset of () by HILB10_3:15;
defpred S1[ Nat, Nat, natural object , Nat, Nat, Nat] means 1 * \$1 = (1 * \$3) * \$2;
A2: for n being Nat
for i1, i2, i3, i4, i5, i6 being Element of n holds { p where p is b1 -element XFinSequence of NAT : S1[p . i1,p . i2,p . i3,p . i4,p . i5,p . i6] } is diophantine Subset of () by HILB10_3:9;
A3: for n being Nat
for i1, i2, i3, i4, i5 being Element of n holds { p where p is b1 -element XFinSequence of NAT : S1[p . i1,p . i2,H1(p . i3,p . i4,p . i5),p . i3,p . i4,p . i5] } is diophantine Subset of () from let i1, i2, i3 be Element of n; :: thesis: { p where p is n -element XFinSequence of NAT : p . i1 = ((a * (p . i2)) + b) * (p . i3) } is diophantine Subset of ()
defpred S2[ XFinSequence of NAT ] means S1[\$1 . i1,\$1 . i3,(a * (\$1 . i2)) + b,\$1 . i3,\$1 . i3,\$1 . i3];
defpred S3[ XFinSequence of NAT ] means \$1 . i1 = ((a * (\$1 . i2)) + b) * (\$1 . i3);
A4: for p being n -element XFinSequence of NAT holds
( S2[p] iff S3[p] ) ;
{ p where p is n -element XFinSequence of NAT : S2[p] } = { q where q is n -element XFinSequence of NAT : S3[q] } from hence { p where p is n -element XFinSequence of NAT : p . i1 = ((a * (p . i2)) + b) * (p . i3) } is diophantine Subset of () by A3; :: thesis: verum