reserve Omega, F for non empty set,
  f for SetSequence of Omega,
  X,A,B for Subset of Omega,
  D for non empty Subset-Family of Omega,
  n,m for Element of NAT,
  h,x,y,z,u,v,Y,I for set;

theorem Th1:
  for f being SetSequence of Omega for x holds x in rng f iff ex n st f.n=x
proof
  let f be SetSequence of Omega;
  let x;
A1: now
    assume x in rng f;
    then consider z being object such that
A2: z in dom f and
A3: x=f.z by FUNCT_1:def 3;
    reconsider n=z as Element of NAT by A2,FUNCT_2:def 1;
    take n;
    thus f.n=x by A3;
  end;
  dom f=NAT by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:def 3;
end;
