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 Th25:
  for f,g being FinSequence of CQC-WFF(Al) st 0 < len f & |- f^<*p*> holds
  |- Ant(f)^g^<*Suc(f)*>^<*p*>
proof
  let f,g be FinSequence of CQC-WFF(Al) such that
A1: 0 < len f and
A2: |- f^<*p*>;
  f is_Subsequence_of Ant(f)^g^<*Suc(f)*> by A1,CALCUL_1:13;
  then Ant(f^<*p*>) is_Subsequence_of Ant(f)^g^<*Suc(f)*> by CALCUL_1:5;
  then
A3: Ant(f^<*p*>) is_Subsequence_of Ant(Ant(f)^g^<*Suc(f)*>^<*p*>)
  by CALCUL_1:5;
  Suc(f^<*p*>) = p by CALCUL_1:5;
  then Suc(f^<*p*>) = Suc(Ant(f)^g^<*Suc(f)*>^<*p*>) by CALCUL_1:5;
  hence thesis by A2,A3,CALCUL_1:36;
end;
