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;
