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 Th37:
  X |= p implies not J,v |= X \/ {'not' p}
proof
  assume
A1: X |= p;
  assume
A2: J,v |= X \/ {'not' p};
  then
A3: J,v |= X by Th35,XBOOLE_1:7;
A4: {'not' p} c= X \/ {'not' p} by XBOOLE_1:7;
  'not' p in {'not' p} by TARSKI:def 1;
  then J,v |= 'not' p by A2,A4,CALCUL_1:def 11;
  then not J,v |= p by VALUAT_1:17;
  hence contradiction by A1,A3,CALCUL_1:def 12;
end;
