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;
