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;

theorem Th17:
  for p holds (CX is negation_faithful & CX is with_examples
  implies (JH,valH(Al) |= p iff CX |- p))
proof
  defpred P[Element of CQC-WFF(Al)] means
  CX is negation_faithful & CX is with_examples
  implies (JH,valH(Al) |= $1 iff CX |- $1);
A1: for p st QuantNbr(p) <= 0 holds P[p] by Th8;
A2: for k st for p st QuantNbr(p) <= k holds P[p] holds
  for p st QuantNbr(p) <= k+1 holds P[p] by Th16;
  thus for p holds P[p] from SUBSTUT2:sch 2(A1,A2);
end;
