reserve Al for QC-alphabet;
reserve a,a1,a2,b,c,d for set,
  X,Y,Z for Subset of CQC-WFF(Al),
  i,k,m,n for Nat,
  p,q for Element of CQC-WFF(Al),
  P for QC-pred_symbol of k,Al,
  ll for CQC-variable_list of k,Al,
  f,f1,f2,g for FinSequence of CQC-WFF(Al);
reserve A for non empty finite Subset of NAT;
reserve C for non empty set;
reserve A for non empty set,
  v for Element of Valuations_in(Al,A),
  J for interpretation of Al,A;
reserve CX for Consistent Subset of CQC-WFF(Al),
  P9 for Element of QC-pred_symbols(Al);
reserve JH for Henkin_interpretation of CX;

theorem
  JH,valH(Al) |= P!ll iff CX |- P!ll
proof
  thus JH,valH(Al) |= P!ll implies CX |- P!ll
  proof
    set rel = JH.P;
    reconsider rel as Element of relations_on HCar(Al);
    assume JH,valH(Al) |= P!ll;
    then Valid(P!ll,JH).valH(Al) = TRUE by VALUAT_1:def 7;
    then ((valH(Al))*'ll) in rel by VALUAT_1:7;
    then ll in rel by Th14;
    then
    ex ll9 being CQC-variable_list of (the_arity_of P), Al st
    ll9 = ll & CX |- P!ll9 by Def5;
    hence thesis;
  end;
  thus CX |- P!ll implies JH,valH(Al) |= P!ll
  proof
    P is (QC-pred_symbol of the_arity_of P, Al) by QC_LANG3:1;
    then
A1: the_arity_of P = k by SUBSTUT2:3;
    assume CX |- P!ll;
    then ll in JH.P by A1,Def5;
    then ((valH(Al))*'ll) in JH.P by Th14;
    then Valid(P!ll,JH).valH(Al) = TRUE by VALUAT_1:7;
    hence thesis by VALUAT_1:def 7;
  end;
end;
