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);
reserve C,D for Subset of CQC-WFF(Al);
reserve JH1 for Henkin_interpretation of CZ,
  J for interpretation of Al, A,
  v for Element of Valuations_in(Al,A);

theorem Th34:
  (Al is countable & still_not-bound_in CX is finite)
   implies ex CZ,JH1 st JH1,valH(Al) |= CX
proof
  assume A1: Al is countable;
  assume still_not-bound_in CX is finite;
  then consider CY such that
A2: CX c= CY and
A3: CY is with_examples by Th31,A1;
  consider CZ such that
A4: CY c= CZ and
A5: CZ is negation_faithful and
A6: CZ is with_examples by A1,A3,Th33;
A7: CX c= CZ by A2,A4;
  set JH1 =the  Henkin_interpretation of CZ;
A8: now
    let p;
    assume p in CX;
    then CZ |- p by A7,Th21;
    hence JH1,valH(Al) |= p by A5,A6,Th17;
  end;
  take CZ,JH1;
  thus thesis by A8,CALCUL_1:def 11;
end;
