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 Th62:
  still_not-bound_in f is finite
proof
  defpred P[object,object] means
ex p st $2 = still_not-bound_in p & p = f.$1;
  consider n being Nat such that
A1: dom f = Seg n by FINSEQ_1:def 2;
  set X = {still_not-bound_in p : ex i st i in dom f & p = f.i};
  consider F1 being Function such that
A2: rng F1 = Seg n and
A3: dom F1 in omega by FINSET_1:def 1;
A4: now
    let b be set;
    assume b in X;
    then ex p st b = still_not-bound_in p & ex i st i in dom f & p = f.i;
    hence b is finite by CQC_SIM1:19;
  end;
A5: for a being object st a in dom f ex b being object st P[a,b]
  proof
    let a be object;
    assume a in dom f;
    then f.a in rng f by FUNCT_1:3;
    then reconsider p = f.a as Element of CQC-WFF(Al);
    take still_not-bound_in p;
    thus thesis;
  end;
  consider F2 being Function such that
A6: dom F2 = dom f &
for b being object st b in dom f holds P[b,F2.b] from CLASSES1:
  sch 1(A5);
  set F = F2*F1;
A7: now
    let b be object;
    assume b in X;
    then consider p such that
A8: b = still_not-bound_in p and
A9: ex i st i in dom f & p = f.i;
    consider i such that
A10: i in dom f and
A11: p = f.i by A9;
    P[i,F2.i] by A6,A10;
    then b in rng F2 by A6,A8,A10,A11,FUNCT_1:3;
    hence b in rng F by A6,A1,A2,RELAT_1:28;
  end;
  now
    let b be object;
    assume b in rng F;
    then b in rng F2 by A6,A1,A2,RELAT_1:28;
    then consider a being object such that
A12: a in dom F2 and
A13: b = F2.a by FUNCT_1:def 3;
    reconsider a as Element of NAT by A6,A12;
    P[a,F2.a] by A6,A12;
    hence b in X by A6,A12,A13;
  end;
  then
A14: rng F = X by A7,TARSKI:2;
  dom F in omega by A6,A1,A2,A3,RELAT_1:27;
  then X is finite by A14,FINSET_1:def 1;
  then union X is finite by A4,FINSET_1:7;
  hence thesis by Th61;
end;
