theorem Th4:
  M is being_a_model_of_ZF implies M |= ZF-axioms
proof
  assume
A1: M is epsilon-transitive & M |= the_axiom_of_pairs & M |=
  the_axiom_of_unions & M |= the_axiom_of_infinity &( M |=
  the_axiom_of_power_sets & for H st {x.0,x.1,x.2} misses Free H holds M |=
  the_axiom_of_substitution_for H);
  let H;
  assume H in ZF-axioms;
  then H = the_axiom_of_extensionality or H = the_axiom_of_pairs or H =
the_axiom_of_unions or H = the_axiom_of_infinity or H = the_axiom_of_power_sets
  or ex H1 being ZF-formula st {x.0,x.1,x.2} misses Free H1 & H =
  the_axiom_of_substitution_for H1 by Def4;
  hence thesis by A1,ZFMODEL1:1;
end;
