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 Th61:
  still_not-bound_in f = union {still_not-bound_in p : ex i st i
  in dom f & p = f.i}
proof
  defpred P[set] means ex p st $1 = still_not-bound_in p & ex i st i in dom f
  & p = f.i;
  set X = {still_not-bound_in p : ex i st i in dom f & p = f.i};
A1: now
    let b be object;
    assume b in union X;
    then consider Y being set such that
A2: b in Y and
A3: Y in X by TARSKI:def 4;
    P[Y] by A3;
    hence b in still_not-bound_in f by A2,Def5;
  end;
  now
    let b be object;
    assume b in still_not-bound_in f;
    then consider i,p such that
A4: i in dom f & p = f.i and
A5: b in still_not-bound_in p by Def5;
    still_not-bound_in p in X by A4;
    hence b in union X by A5,TARSKI:def 4;
  end;
  hence thesis by A1,TARSKI:2;
end;
