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 Th11:
  J,v |= 'not' Ex(x,'not' p) iff J,v |= All(x,p)
proof
A1: not J,v |= Ex(x,'not' p) iff for a holds not J,v.(x|a) |= 'not' p by Th9;
A2: (for a holds not J,v.(x|a) |= 'not' p) implies for a holds J,v.(x|a) |= p
  proof
    assume
A3: for a holds not J,v.(x|a) |= 'not' p;
    let a;
    not J,v.(x|a) |= 'not' p by A3;
    hence thesis by VALUAT_1:17;
  end;
  (for a holds J,v.(x|a) |= p) implies for a holds not J,v.(x|a) |= 'not' p
     by VALUAT_1:17;
  hence thesis by A1,A2,SUBLEMMA:50,VALUAT_1:17;
end;
