reserve H,S for ZF-formula,
  x for Variable,
  X,Y for set,
  i for Element of NAT,
  e,u for set;
reserve M,M1,M2 for non empty set,
  f for Function,
  v1 for Function of VAR,M1,
  v2 for Function of VAR,M2,
  F,F1,F2 for Subset of WFF,
  W for Universe,
  a,b,c for Ordinal of W,
  A,B,C for Ordinal,
  L for DOMAIN-Sequence of W,
  va for Function of VAR,L.a,
  phi,xi for Ordinal-Sequence of W;
reserve psi for Ordinal-Sequence;

theorem
  omega in W & (for a,b st a in b holds L.a c= L.b) & (for a st a <> {}
  & a is limit_ordinal holds L.a = Union (L|a)) implies ex phi st phi is
  increasing & phi is continuous & for a st phi.a = a & {} <> a holds L.a <==>
  Union L
proof
  assume ( omega in W & for a,b st a in b holds L.a c= L.b )& for a st a <>
  {} & a is limit_ordinal holds L.a = Union (L|a);
  then consider phi such that
A1: phi is increasing & phi is continuous and
A2: for a st phi.a = a & {} <> a holds L.a is_elementary_subsystem_of
  Union L by Th30;
  take phi;
  thus phi is increasing & phi is continuous by A1;
  let a;
  assume phi.a = a & {} <> a;
  hence thesis by A2,Th9;
end;
