let n be Nat; :: thesis: for a, b, c being Integer
for i1, i2, i3 being Element of n holds { p where p is n -element XFinSequence of NAT : a * (p . i1) < (b * (p . i2)) + (c * (p . i3)) } is diophantine Subset of (n -xtuples_of NAT)
let a, b, c be Integer; :: thesis: for i1, i2, i3 being Element of n holds { p where p is n -element XFinSequence of NAT : a * (p . i1) < (b * (p . i2)) + (c * (p . i3)) } is diophantine Subset of (n -xtuples_of NAT)
let i1, i2, i3 be Element of n; :: thesis: { p where p is n -element XFinSequence of NAT : a * (p . i1) < (b * (p . i2)) + (c * (p . i3)) } is diophantine Subset of (n -xtuples_of NAT)
defpred S₁[ Nat, Nat, Integer] means (a * $1) + 0 < $3;
deffunc H₁( Nat, Nat, Nat) -> set = ((b * $2) + (c * $3)) + 0;
defpred S₂[ XFinSequence of NAT ] means (a * ($1 . i1)) + 0 < ((b * ($1 . i2)) + (c * ($1 . i3))) + 0;
defpred S₃[ XFinSequence of NAT ] means a * ($1 . i1) < (b * ($1 . i2)) + (c * ($1 . i3));
A1: for n being Nat
for i1, i2, i3 being Element of n
for d being Integer holds { p where p is b₁ -element XFinSequence of NAT : S₁[p . i1,p . i2,d * (p . i3)] } is diophantine Subset of (n -xtuples_of NAT) by Th7;
A2: for n being Nat
for i1, i2, i3, i4 being Element of n
for d being Integer holds { p where p is b₁ -element XFinSequence of NAT : H₁(p . i1,p . i2,p . i3) = d * (p . i4) } is diophantine Subset of (n -xtuples_of NAT) by Th11;
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)] } is diophantine Subset of (n -xtuples_of NAT) from HILB10_3:sch 5(A1, A2);
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 HILB10_3:sch 2(A4);
hence { p where p is n -element XFinSequence of NAT : a * (p . i1) < (b * (p . i2)) + (c * (p . i3)) } is diophantine Subset of (n -xtuples_of NAT) by A3; :: thesis: verum