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 Th6:
  ex S st Free S = {} & for M holds M |= S iff M |= H
proof
  defpred P[Nat] means for H st card Free H = $1 ex S st Free S =
  {} & for M holds M |= S iff M |= H;
A1: for i being Nat holds P[i] implies P[i+1]
  proof let i be Nat;
    assume
A2: P[i];
    let H;
    set e = the Element of Free H;
    assume
A3: card Free H = i+1;
    then
A4: Free H <> {};
    then reconsider x = e as Variable by TARSKI:def 3;
A5: {x} c= Free H by A4,ZFMISC_1:31;
A6: Free All(x,H) = Free H \ {x} by ZF_LANG1:62;
    x in {x} by ZFMISC_1:31;
    then
A7: not x in Free All(x,H) by A6,XBOOLE_0:def 5;
    Free All(x,H) \/ {x} = Free H \/ {x} by A6,XBOOLE_1:39;
    then Free All(x,H) \/ {x} = Free H by A5,XBOOLE_1:12;
    then card Free All(x,H) + 1 = card Free H by A7,CARD_2:41;
    then consider S such that
A8: Free S = {} and
A9: for M holds M |= S iff M |= All(x,H) by A2,A3,XCMPLX_1:2;
    take S;
    thus Free S = {} by A8;
    let M;
    M |= H iff M |= All(x,H) by ZF_MODEL:23;
    hence thesis by A9;
  end;
A10: card Free H = card Free H;
A11: P[0]
  proof
    let H;
    assume
A12: card Free H = 0;
    take H;
    thus thesis by A12;
  end;
  for i being Nat holds P[i] from NAT_1:sch 2(A11,A1);
  hence thesis by A10;
end;
