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;
reserve L for PATH of q,p,
  F1,F3 for QC-formula of Al,
  a for set;
reserve C,D for Element of [:CQC-WFF(Al),bool bound_QC-variables(Al):];
reserve K,L for Subset of bound_QC-variables(Al);

theorem Th30:
  for f being FinSequence of CQC-WFF(Al) holds
  still_not-bound_in rng f = still_not-bound_in f
proof
  let f be FinSequence of CQC-WFF(Al);
  set A = {still_not-bound_in p : p in rng f};
A1: now
    let a be object;
    assume a in still_not-bound_in rng f;
    then consider b being set such that
A2: a in b and
A3: b in A by TARSKI:def 4;
    consider p such that
A4: b = still_not-bound_in p and
A5: p in rng f by A3;
    ex c being object st ( c in dom f)&( f.c = p) by A5,FUNCT_1:def 3;
    hence a in still_not-bound_in f by A2,A4,CALCUL_1:def 5;
  end;
  now
    let a be object;
    assume a in still_not-bound_in f;
    then consider i being Nat,q such that
A6: i in dom f and
A7: q = f.i and
A8: a in still_not-bound_in q by CALCUL_1:def 5;
    q in rng f by A6,A7,FUNCT_1:def 3;
    then still_not-bound_in q in A;
    hence a in still_not-bound_in rng f by A8,TARSKI:def 4;
  end;
  hence thesis by A1,TARSKI:2;
end;
