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 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;
