reserve a,b,k,m,n,s for Nat;
reserve c,c1,c2,c3 for Complex;
reserve i,j,z for Integer;
reserve p for Prime;
reserve x for object;
reserve f,g for complex-valued FinSequence;

theorem Th85:
  rng exampleSierpinski196 c=
  { [x,y,z,t] where x,y,z,t is Integer: x+y = z*t & z+t = x*y }
  proof
    let y be object;
    assume y in rng f;
    then consider k being object such that
A1: k in dom f and
A2: f.k = y by FUNCT_1:def 3;
    reconsider k as Element of NAT by A1;
    X(k)+Y(k) = Z(k)*T(k) & Z(k)+T(k) = X(k)*Y(k);
    then F(k) in { [x,y,z,t] where x,y,z,t is Integer: x+y = z*t & z+t = x*y };
    hence thesis by A2,Def16;
  end;
