reserve Al for QC-alphabet;
reserve a,b,c,d for object,
  i,k,n for Nat,
  p,q for Element of CQC-WFF(Al),
  x,y,y1 for bound_QC-variable of Al,
  A for non empty set,
  J for interpretation of Al,A,
  v,w for Element of Valuations_in(Al,A),
  f,g for Function,
  P,P9 for QC-pred_symbol of k,Al,
  ll,ll9 for CQC-variable_list of k,Al,
  l1 for FinSequence of QC-variables(Al),
  Sub,Sub9,Sub1 for CQC_Substitution of Al,
  S,S9,S1,S2 for Element of CQC-Sub-WFF(Al),
  s for QC-symbol of Al;
reserve vS,vS1,vS2 for Val_Sub of A,Al;
reserve B for Element of [:QC-Sub-WFF(Al),bound_QC-variables(Al):],
  SQ for second_Q_comp of B;
reserve B for CQC-WFF-like Element of [:QC-Sub-WFF(Al),
  bound_QC-variables(Al):],
  xSQ for second_Q_comp of [S,x],
  SQ for second_Q_comp of B;
reserve B1 for Element of [:QC-Sub-WFF(Al),bound_QC-variables(Al):];
reserve SQ1 for second_Q_comp of B1;

theorem Th44:
  still_not-bound_in (p) c= Bound_Vars(p) implies
  still_not-bound_in ('not' p) c= Bound_Vars('not' p)
proof
  'not' p is negative by QC_LANG1:def 19;
  then Bound_Vars('not' p) = Bound_Vars(the_argument_of ('not' p)) by
SUBSTUT1:4;
  then
A1: Bound_Vars('not' p) = Bound_Vars(p) by QC_LANG2:1;
  assume still_not-bound_in (p) c= Bound_Vars(p);
  hence thesis by A1,QC_LANG3:7;
end;
