reserve Al for QC-alphabet;
reserve a,b,c,d for object,
  i,j,k,m,n for Nat,
  p,q,r for Element of CQC-WFF(Al),
  x,y,y0 for bound_QC-variable of Al,
  X for Subset of CQC-WFF(Al),
  A for non empty set,
  J for interpretation of Al,A,
  v,w for Element of Valuations_in(Al,A),
  Sub for CQC_Substitution of Al,
  f,f1,g,h,h1 for FinSequence of CQC-WFF(Al);
reserve fin,fin1 for FinSequence;
reserve PR,PR1 for FinSequence of [:set_of_CQC-WFF-seq(Al),Proof_Step_Kinds:];
reserve a for Element of A;

theorem Th56:
  ('not' p).(x,y) = 'not' (p.(x,y))
proof
  set S = [p,Sbst(x,y)];
  S`1 = p & S`2 = Sbst(x,y);
  then ('not' p).(x,y) = CQC_Sub(['not' p,Sbst(x,y)]) & Sub_not S = ['not' p,
  Sbst(x,y)] by SUBSTUT1:def 20,SUBSTUT2:def 1;
  then ('not' p).(x,y) = 'not' CQC_Sub(S) by SUBSTUT1:29;
  hence thesis by SUBSTUT2:def 1;
end;
