theorem Th9:
  J,v |= Ex(x,p) iff ex a st J,v.(x|a) |= p
proof
A1: J,v |= 'not' All(x,'not' p) iff not J,v |= All(x,'not' p) by VALUAT_1:17;
A2: (ex a st not J,v.(x|a) |= 'not' p) implies ex a st J,v.(x|a) |= p
           by VALUAT_1:17;
  (ex a st J,v.(x|a) |= p) implies ex a st not J,v.(x|a) |= 'not' p
  proof
    given a such that
A3: J,v.(x|a) |= p;
    take a;
    thus thesis by A3,VALUAT_1:17;
  end;
  hence thesis by A1,A2,QC_LANG2:def 5,SUBLEMMA:50;
end;
