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 Th9:
  M1 is_elementary_subsystem_of M2 implies M1 <==> M2
proof
  assume that
  M1 c= M2 and
A1: for H for v being Function of VAR,M1 holds M1,v |= H iff M2,M2!v |= H;
  let H such that
A2: Free H = {};
  thus M1 |= H implies M2 |= H
  proof
    assume
A3: M1,v1 |= H;
    set v1 = the Function of VAR,M1;
    M1,v1 |= H by A3;
    then
A4: M2,M2!v1 |= H by A1;
    let v2;
    for x st x in Free H holds v2.x = (M2!v1).x by A2;
    hence thesis by A4,ZF_LANG1:75;
  end;
  assume
A5: M2,v2 |= H;
  let v1;
  M2,M2!v1 |= H by A5;
  hence thesis by A1;
end;
