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
  ( Al is countable &
  still_not-bound_in X is finite & X |= p ) implies X |- p
proof
  assume
A1: Al is countable;
  assume
A2: still_not-bound_in X is finite;
  assume
A3: X |= p;
  assume
A4: not X |- p;
  reconsider Y = X \/ {'not' p} as Subset of CQC-WFF(Al);
A5: still_not-bound_in Y is finite by A2,Th36;
  Y is Consistent by A4,HENMODEL:9;
  then ex CZ,JH1 st ( JH1,valH(Al) |= Y) by A1,A5,Th34;
  hence contradiction by A3,Th37;
end;
