let n be Nat; :: thesis: for a, b 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)) * (p . i3) } is diophantine Subset of (n -xtuples_of NAT)
let a, b 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)) * (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)) * (p . i3) } is diophantine Subset of (n -xtuples_of NAT)
set F = F_Real ;
reconsider ar = a, br = b as integer Element of F_Real by XREAL_0:def 1;
set D = { p where p is n -element XFinSequence of NAT : a * (p . i1) = (b * (p . i2)) * (p . i3) } ;
{ p where p is n -element XFinSequence of NAT : a * (p . i1) = (b * (p . i2)) * (p . i3) } c= n -xtuples_of NAT

proof

let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in { p where p is n -element XFinSequence of NAT : a * (p . i1) = (b * (p . i2)) * (p . i3) } or y in n -xtuples_of NAT )
assume y in { p where p is n -element XFinSequence of NAT : a * (p . i1) = (b * (p . i2)) * (p . i3) } ; :: thesis: y in n -xtuples_of NAT
then ex p being n -element XFinSequence of NAT st
( y = p & a * (p . i1) = (b * (p . i2)) * (p . i3) ) ;
hence y in n -xtuples_of NAT by HILB10_2:def 5; :: thesis: verum

end;

then reconsider D = { p where p is n -element XFinSequence of NAT : a * (p . i1) = (b * (p . i2)) * (p . i3) } as Subset of (n -xtuples_of NAT) ;

per cases ( n = 0 or n <> 0 ) ;

suppose n = 0 ; :: thesis: { p where p is n -element XFinSequence of NAT : a * (p . i1) = (b * (p . i2)) * (p . i3) } is diophantine Subset of (n -xtuples_of NAT)

then D is diophantine Subset of (n -xtuples_of NAT) ;
hence { p where p is n -element XFinSequence of NAT : a * (p . i1) = (b * (p . i2)) * (p . i3) } is diophantine Subset of (n -xtuples_of NAT) ; :: thesis: verum

end;

suppose A1: n <> 0 ; :: thesis: { p where p is n -element XFinSequence of NAT : a * (p . i1) = (b * (p . i2)) * (p . i3) } is diophantine Subset of (n -xtuples_of NAT)

set Q = (ar * (1_1 (i1,F_Real))) - (br * ((1_1 (i2,F_Real)) *' (1_1 (i3,F_Real))));
reconsider Q = (ar * (1_1 (i1,F_Real))) - (br * ((1_1 (i2,F_Real)) *' (1_1 (i3,F_Real)))) as INT -valued Polynomial of (n + 0),F_Real ;
for s being object holds
( s in D iff ex x being n -element XFinSequence of NAT ex y being 0 -element XFinSequence of NAT st
( s = x & eval (Q,(@ (x ^ y))) = 0 ) )

proof

let s be object ; :: thesis: ( s in D iff ex x being n -element XFinSequence of NAT ex y being 0 -element XFinSequence of NAT st
( s = x & eval (Q,(@ (x ^ y))) = 0 ) )
thus ( s in D implies ex x being n -element XFinSequence of NAT ex y being 0 -element XFinSequence of NAT st
( s = x & eval (Q,(@ (x ^ y))) = 0 ) ) :: thesis: ( ex x being n -element XFinSequence of NAT ex y being 0 -element XFinSequence of NAT st
( s = x & eval (Q,(@ (x ^ y))) = 0 ) implies s in D )

proof

assume s in D ; :: thesis: ex x being n -element XFinSequence of NAT ex y being 0 -element XFinSequence of NAT st
( s = x & eval (Q,(@ (x ^ y))) = 0 )
then consider p being n -element XFinSequence of NAT such that
A2: ( s = p & a * (p . i1) = (b * (p . i2)) * (p . i3) ) ;
reconsider Z = <%> NAT as 0 -element XFinSequence of NAT ;
take p ; :: thesis: ex y being 0 -element XFinSequence of NAT st
( s = p & eval (Q,(@ (p ^ y))) = 0 )
take Z ; :: thesis: ( s = p & eval (Q,(@ (p ^ Z))) = 0 )
set pZ = p ^ Z;
A3: ( eval ((1_1 (i1,F_Real)),(@ p)) = p . i1 & eval ((1_1 (i2,F_Real)),(@ p)) = p . i2 & eval ((1_1 (i3,F_Real)),(@ p)) = p . i3 ) by A1, Th1;
A4: eval ((ar * (1_1 (i1,F_Real))),(@ p)) = ar * (eval ((1_1 (i1,F_Real)),(@ p))) by POLYNOM7:29
.= a * (p . i1) by A1, Th1 ;
A5: eval ((br * ((1_1 (i2,F_Real)) *' (1_1 (i3,F_Real)))),(@ p)) = br * (eval (((1_1 (i2,F_Real)) *' (1_1 (i3,F_Real))),(@ p))) by POLYNOM7:29
.= br * ((eval ((1_1 (i2,F_Real)),(@ p))) * (eval ((1_1 (i3,F_Real)),(@ p)))) by POLYNOM2:25
.= b * ((p . i2) * (p . i3)) by A3 ;
eval (Q,(@ (p ^ Z))) = (eval ((ar * (1_1 (i1,F_Real))),(@ p))) - (eval ((br * ((1_1 (i2,F_Real)) *' (1_1 (i3,F_Real)))),(@ p))) by POLYNOM2:24
.= 0 by A2, A4, A5 ;
hence ( s = p & eval (Q,(@ (p ^ Z))) = 0 ) by A2; :: thesis: verum

end;

given p being n -element XFinSequence of NAT , Z being 0 -element XFinSequence of NAT such that A6: ( s = p & eval (Q,(@ (p ^ Z))) = 0 ) ; :: thesis: s in D
A7: ( eval ((1_1 (i1,F_Real)),(@ p)) = p . i1 & eval ((1_1 (i2,F_Real)),(@ p)) = p . i2 & eval ((1_1 (i3,F_Real)),(@ p)) = p . i3 ) by A1, Th1;
A8: eval ((br * ((1_1 (i2,F_Real)) *' (1_1 (i3,F_Real)))),(@ p)) = br * (eval (((1_1 (i2,F_Real)) *' (1_1 (i3,F_Real))),(@ p))) by POLYNOM7:29
.= br * ((eval ((1_1 (i2,F_Real)),(@ p))) * (eval ((1_1 (i3,F_Real)),(@ p)))) by POLYNOM2:25
.= b * ((p . i2) * (p . i3)) by A7 ;
A9: eval ((ar * (1_1 (i1,F_Real))),(@ p)) = ar * (eval ((1_1 (i1,F_Real)),(@ p))) by POLYNOM7:29
.= a * (p . i1) by A1, Th1 ;
set P = p ^ Z;
eval (Q,(@ (p ^ Z))) = (eval ((ar * (1_1 (i1,F_Real))),(@ p))) - (eval ((br * ((1_1 (i2,F_Real)) *' (1_1 (i3,F_Real)))),(@ p))) by POLYNOM2:24
.= (a * (p . i1)) - (b * ((p . i2) * (p . i3))) by A8, A9 ;
then a * (p . i1) = (b * (p . i2)) * (p . i3) by A6;
hence s in D by A6; :: thesis: verum

end;

then D is diophantine ;
hence { p where p is n -element XFinSequence of NAT : a * (p . i1) = (b * (p . i2)) * (p . i3) } is diophantine Subset of (n -xtuples_of NAT) ; :: thesis: verum

end;

end;