set A = { [x,y] where x, y is Nat : x,y satisfy_Sierpinski_problem_35 } ;
deffunc H₁( Nat) -> object = [((36 * $1) + 14),(((12 * $1) + 5) * ((18 * $1) + 7))];
consider f being ManySortedSet of NAT such that
A1: for d being Element of NAT holds f . d = H₁(d) from PBOOLE:sch 5();
A2: dom f = NAT by PARTFUN1:def 2;
A3: rng f c= { [x,y] where x, y is Nat : x,y satisfy_Sierpinski_problem_35 }

proof

let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in rng f or y in { [x,y] where x, y is Nat : x,y satisfy_Sierpinski_problem_35 } )
assume y in rng f ; :: thesis: y in { [x,y] where x, y is Nat : x,y satisfy_Sierpinski_problem_35 }
then consider k being object such that
A4: k in dom f and
A5: f . k = y by FUNCT_1:def 3;
reconsider k = k as Element of NAT by A4, PARTFUN1:def 2;
(36 * k) + 14,((12 * k) + 5) * ((18 * k) + 7) satisfy_Sierpinski_problem_35 by Th15;
then H₁(k) in { [x,y] where x, y is Nat : x,y satisfy_Sierpinski_problem_35 } ;
hence y in { [x,y] where x, y is Nat : x,y satisfy_Sierpinski_problem_35 } by A1, A5; :: thesis: verum

end;

f is one-to-one

proof

let x1, x2 be object ; :: according to FUNCT_1:def 4 :: thesis: ( not x1 in dom f or not x2 in dom f or not f . x1 = f . x2 or x1 = x2 )
assume that
A6: ( x1 in dom f & x2 in dom f ) and
A7: f . x1 = f . x2 ; :: thesis: x1 = x2
reconsider x1 = x1, x2 = x2 as Element of NAT by A6, PARTFUN1:def 2;
( f . x1 = H₁(x1) & f . x2 = H₁(x2) ) by A1;
then (36 * x1) + 14 = (36 * x2) + 14 by A7, XTUPLE_0:1;
hence x1 = x2 ; :: thesis: verum

end;

hence { [x,y] where x, y is Nat : x,y satisfy_Sierpinski_problem_35 } is infinite by A2, A3, CARD_1:59; :: thesis: verum