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;

theorem Th8:
  for p st QuantNbr(p) <= 0 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 r,s,x,k for l being CQC-variable_list of k,Al
  for P being QC-pred_symbol of k,Al holds P[VERUM(Al)] & P[P!l] &
  (P[r] implies P['not' r]) & (P[r] & P[s] implies P[r '&' s])
  by Def1,Th6,HENMODEL:16,17,def 2,VALUAT_1:17,18;
A2: for p st QuantNbr(p) = 0 holds P[p] from SUBSTUT2:sch 3(A1);
  now
    let p;
    assume QuantNbr(p) <= 0;
    then QuantNbr(p) = 0 by NAT_1:2;
    hence P[p] by A2;
  end;
  hence thesis;
end;
