theorem
  M1 <==> M2 iff for H holds M1 |= H iff M2 |= H
proof
  thus M1 <==> M2 implies for H holds M1 |= H iff M2 |= H
  proof
    assume
A1: for H st Free H = {} holds M1 |= H iff M2 |= H;
    let H;
    consider S such that
A2: Free S = {} and
A3: for M holds M |= S iff M |= H by Th6;
    M1 |= S iff M2 |= S by A1,A2;
    hence thesis by A3;
  end;
  assume
A4: for H holds M1 |= H iff M2 |= H;
  let H;
  thus thesis by A4;
end;
