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;
