reserve Al for QC-alphabet;
reserve a,b,c,d for object,
  i,j,k,m,n for Nat,
  p,q,r for Element of CQC-WFF(Al),
  x,y,y0 for bound_QC-variable of Al,
  X for Subset of CQC-WFF(Al),
  A for non empty set,
  J for interpretation of Al,A,
  v,w for Element of Valuations_in(Al,A),
  Sub for CQC_Substitution of Al,
  f,f1,g,h,h1 for FinSequence of CQC-WFF(Al);
reserve fin,fin1 for FinSequence;
reserve PR,PR1 for FinSequence of [:set_of_CQC-WFF-seq(Al),Proof_Step_Kinds:];
reserve a for Element of A;

theorem
  |- f^<*p.(x,y)*>^<*q*> & not y in still_not-bound_in (f^<*Ex(x,p)*>^<*
  q*>) implies |- f^<*Ex(x,p)*>^<*q*>
proof
  assume that
A1: |- f^<*p.(x,y)*>^<*q*> and
A2: not y in still_not-bound_in (f^<*Ex(x,p)*>^<*q*>);
  set f1 = f^<*'not' q*>^<*('not' p).(x,y)*>;
  |- f^<*'not' q*>^<*'not' (p.(x,y))*> by A1,Th46;
  then
A3: |- f^<*'not' q*>^<*('not' p).(x,y)*> by Th56;
A4: not y in still_not-bound_in (f^<*Ex(x,p)*>) \/ still_not-bound_in <*q*>
  by A2,Th58;
  then not y in still_not-bound_in (f^<*Ex(x,p)*>) by XBOOLE_0:def 3;
  then
A5: not y in still_not-bound_in f \/ still_not-bound_in <*Ex(x,p)*> by Th58;
  then not y in still_not-bound_in <*Ex(x,p)*> by XBOOLE_0:def 3;
  then not y in still_not-bound_in Ex(x,p) by Th59;
  then not y in still_not-bound_in p \ {x} by QC_LANG3:19;
  then not y in still_not-bound_in 'not' p \ {x} by QC_LANG3:7;
  then
A6: not y in still_not-bound_in All(x,'not' p) by QC_LANG3:12;
  not y in still_not-bound_in <*q*> by A4,XBOOLE_0:def 3;
  then not y in still_not-bound_in q by Th59;
  then not y in still_not-bound_in 'not' q by QC_LANG3:7;
  then
A7: not y in still_not-bound_in <*'not' q*> by Th59;
  not y in still_not-bound_in f by A5,XBOOLE_0:def 3;
  then not y in still_not-bound_in f \/ still_not-bound_in <*'not' q*> by A7,
XBOOLE_0:def 3;
  then not y in still_not-bound_in (f^<*'not' q*>) by Th58;
  then
A8: not y in still_not-bound_in Ant(f1) by Th5;
  Suc(f1) = ('not' p).(x,y) by Th5;
  then |- Ant(f1)^<*All(x,'not' p)*> by A3,A8,A6,Th43;
  then |- f^<*'not' q*>^<*All(x,'not' p)*> by Th5;
  then |- f^<*'not' All(x,'not' p)*>^<*q*> by Th48;
  hence thesis by QC_LANG2:def 5;
end;
