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;

theorem Th10:
  M1 is being_a_model_of_ZF & M1 <==> M2 & M2 is
  epsilon-transitive implies M2 is being_a_model_of_ZF
proof
  assume that
A1: M1 is being_a_model_of_ZF and
A2: M1 <==> M2;
  M1 |= ZF-axioms by A1,Th4;
  then M2 |= ZF-axioms by A2,Th8;
  hence thesis by Th5;
end;
