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 Th27:
  for v,w holds v|still_not-bound_in f = w|still_not-bound_in f
  implies (J,v |= f implies J,w |= f)
proof
  let v,w such that
A1: v|still_not-bound_in f = w|still_not-bound_in f;
  assume J,v |= f;
  then
A2: J,v |= rng(f);
  let p such that
A3: p in rng(f);
  ex i being Nat st i in dom f & p = f.i by A3,FINSEQ_2:10;
  then
  for b being object st b in still_not-bound_in p
holds b in still_not-bound_in f by Def5;
  then still_not-bound_in p c= still_not-bound_in f;
  then
A4: v|still_not-bound_in p = w|still_not-bound_in p by A1,RELAT_1:153;
  J,v |= p by A2,A3;
  hence thesis by A4,SUBLEMMA:68;
end;
