reserve Al for QC-alphabet;
reserve b,c,d for set,
  X,Y for Subset of CQC-WFF(Al),
  i,j,k,m,n for Nat,
  p,p1,q,r,s,s1 for Element of CQC-WFF(Al),
  x,x1,x2,y,y1 for bound_QC-variable of Al,
  A for non empty set,
  J for interpretation of Al, A,
  v for Element of Valuations_in(Al,A),
  f1,f2 for FinSequence of CQC-WFF(Al),
  CX,CY,CZ for Consistent Subset of CQC-WFF(Al),
  JH for Henkin_interpretation of CX,
  a for Element of A,
  t,u for QC-symbol of Al;

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;
