reserve k,n,m for Nat,
  A,B,C for Ordinal,
  X for set,
  x,y,z for object;
reserve f,g,h,fx for Function,
  K,M,N for Cardinal,
  phi,psi for
  Ordinal-Sequence;

theorem Th2:
  y in Union f iff ex x st x in dom f & y in f.x
proof
  thus y in Union f implies ex x st x in dom f & y in f.x
  proof
    assume y in Union f;
    then consider X such that
A1: y in X and
A2: X in rng f by TARSKI:def 4;
    consider x being object such that
A3: x in dom f & X = f.x by A2,FUNCT_1:def 3;
    take x;
    thus thesis by A1,A3;
  end;
  given x such that
A4: x in dom f and
A5: y in f.x;
  f.x in rng f by A4,FUNCT_1:def 3;
  hence thesis by A5,TARSKI:def 4;
end;
