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;
reserve z for Complex;

theorem
  { [x,y,z,t] where x,y,z,t is Integer: x/y + y/z + z/t + t/x = 1 } is infinite
  proof
    set G2 = GreaterOrEqualsNumbers(2);
    set A = { [x,y,z,t] where x,y,z,t is Integer: x/y + y/z + z/t + t/x = 1 };
    deffunc x(Nat) = -$1^2;
    deffunc y(Nat) = $1^2*($1^2-1);
    deffunc z(Nat) = ($1^2-1)^2;
    deffunc t(Nat) = -$1*($1^2-1);
    deffunc F(Element of G2) = [x($1),y($1),z($1),t($1)];
    consider f being ManySortedSet of G2 such that
A1: for d being Element of G2 holds f.d = F(d) from PBOOLE:sch 5;
A2: dom f = G2 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 G2 by A4,PARTFUN1:def 2;
      set q = k^2-1;
A6:   x(k)/y(k) = ((-1)*k^2) / ((k^2-1)*k^2)
      .= (-1)/q by XCMPLX_1:91;
A7:   z(k)/t(k) = q^2 / ((-k)*q)
      .= q/-k by XCMPLX_1:91
      .= -q/k by XCMPLX_1:188;
A8:   t(k)/x(k) = (k*q)/k^2 by XCMPLX_1:191
      .= q/k by XCMPLX_1:91;
      k^2/q - 1/q = q/q
      .= 1 by XCMPLX_1:60;
      then x(k)/y(k) + y(k)/z(k) + z(k)/t(k) + t(k)/x(k) = 1
      by A6,A7,A8,XCMPLX_1:91;
      then F(k) in A;
      hence thesis by A1,A5;
    end;
    f is one-to-one
    proof
      let x1,x2 be object such that
A9:   x1 in dom f & x2 in dom f and
A10:  f.x1 = f.x2;
      reconsider x1,x2 as Element of G2 by A9,PARTFUN1:def 2;
      f.x1 = F(x1) & f.x2 = F(x2) by A1;
      then x(x1) = x(x2) by A10,XTUPLE_0:5;
      then x1 = x2 or x1 = -x2 by SQUARE_1:40;
      hence thesis;
    end;
    hence thesis by A2,A3,CARD_1:59;
  end;
