reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;

theorem
  { [x,y] where x,y is positive Rational: 2*x|^3 + x*y - 7 = 0 } is infinite
  proof
    deffunc R(Rational,Rational) = 2*$1|^3 + $1*$2 - 7;
    set A = { [x,y] where x,y is positive Rational: R(x,y) = 0 };
    deffunc p1(Nat) = 7/$1;
    deffunc p2(Nat) = $1-98/$1^2;
    deffunc F(Nat) = [ p1($1) , p2($1) ];
    set D = NAT \ {0,1,2,3,4};
    reconsider D as infinite natural-membered set;
    consider f being ManySortedSet of D such that
A1: for d being Element of D holds f.d = F(d) from PBOOLE:sch 5;
A2: dom f = D by PARTFUN1:def 2;
A3: rng f c= A
    proof
      let y be object;
      assume y in rng f;
      then consider k being object such that
A4:   k in dom f and
A5:   f.k = y by FUNCT_1:def 3;
      reconsider k as Element of D by A4,PARTFUN1:def 2;
A6:   not k in {0,1,2,3,4} by XBOOLE_0:def 5;
      then
A7:   k <> 0 by ENUMSET1:def 3;
      then
A8:   R(p1(k),p2(k)) = 0 by Th96;
      now
        assume
A9:     p2(k) <= 0;
        (k-98/k^2)*k^2 = k*k^2-98/k^2*k^2
        .= k*k*k-98 by A7,XCMPLX_1:87;
        then k*k*k-98+98 <= 0+98 by A9,XREAL_1:6;
        then k|^3 <= 98 by POLYEQ_5:2;
        then k <= 4 by Th97;
        then k = 0 or ... or k = 4;
        hence contradiction by A6,ENUMSET1:def 3;
      end;
      then F(k) in A by A7,A8;
      hence thesis by A1,A5;
    end;
    f is one-to-one
    proof
      let x1,x2 be object such that
A10:  x1 in dom f & x2 in dom f and
A11:  f.x1 = f.x2;
      reconsider x1,x2 as Element of D by A10,PARTFUN1:def 2;
      f.x1 = F(x1) & f.x2 = F(x2) by A1;
      then p1(x1) = p1(x2) by A11,XTUPLE_0:1;
      hence thesis by XCMPLX_1:201;
    end;
    hence thesis by A2,A3,CARD_1:59;
  end;
