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 Th44:
  |- f & |- Ant(f)^<*'not' Suc(f)*> implies |- Ant(f)^<*p*>
proof
  assume that
A1: |- f and
A2: |- Ant(f)^<*'not' Suc(f)*>;
  set f1 = Ant(f)^<*'not' p*>^<*Suc(f)*>;
  Ant(f1) = Ant(f)^<*'not' p*> & Suc(f) = Suc(f1) by Th5;
  then
A3: |- f1 by A1,Th8,Th36;
  set f3 = Ant(f)^<*'not' p*>^<*'not' Suc(f)*>;
  set f2 = Ant(f)^<*'not' Suc(f)*>;
  Suc(f2) = 'not' Suc(f) by Th5;
  then
A4: Suc(f2) = Suc(f3) by Th5;
  Ant(f3) = Ant(f)^<*'not' p*> & Ant(f2) = Ant(f) by Th5;
  then
A5: |- f3 by A2,A4,Th8,Th36;
  Suc(f1) = Suc(f) by Th5;
  then
A6: 'not' Suc(f1) = Suc(f3) by Th5;
A7: 1< len f1 by Th9;
A8: Ant(f1) = Ant(f)^<*'not' p*> by Th5;
  then Ant(f1) = Ant(f3) & Suc(Ant(f1)) = 'not' p by Th5;
  then |- Ant(Ant(f1))^<*p*> by A3,A5,A6,A7,Th38;
  hence thesis by A8,Th5;
end;
