theorem Th12:
  X |- 'not' Ex(x,'not' p) iff X |- All(x,p)
proof
  thus X |- 'not' Ex(x,'not' p) implies X |- All(x,p)
  proof
    assume X |- 'not' Ex(x,'not' p);
    then consider f1 such that
A1: rng f1 c= X and
A2: |- f1^<*'not' Ex(x,'not' p)*> by HENMODEL:def 1;
    |- f1^<*All(x,p)*> by A2,CALCUL_1:68;
    hence thesis by A1,HENMODEL:def 1;
  end;
  thus X |- All(x,p) implies X |- 'not' Ex(x,'not' p)
  proof
    assume X |- All(x,p);
    then consider f1 such that
A3: rng f1 c= X and
A4: |- f1^<*All(x,p)*> by HENMODEL:def 1;
    |- f1^<*'not' Ex(x,'not' p)*> by A4,CALCUL_1:68;
    hence thesis by A3,HENMODEL:def 1;
  end;
end;
