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:];

theorem Th15:
  Suc(f) is_tail_of Ant(f) implies Ant(f) |= Suc(f)
proof
  assume Suc(f) is_tail_of Ant(f);
  then ex i st i in dom Ant(f) & (Ant(f)).i = Suc(f) by Lm1;
  then
A1: Suc(f) in rng(Ant(f)) by FUNCT_1:3;
  let A,J,v;
  assume J,v |= rng(Ant(f));
  hence thesis by A1;
end;
