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 Th45:
  1 <= len f & |- f & |- f^<*p*> implies |- Ant(f)^<*p*>
proof
  assume that
A1: 1 <= len f and
A2: |- f and
A3: |- f^<*p*>;
  set f2 = Ant(f)^<*'not' Suc(f)*>^<*'not' Suc(f)*>;
  set f1 = Ant(f)^<*'not' Suc(f)*>^<*Suc(f)*>;
A4: Ant(f2) = Ant(f)^<*'not' Suc(f)*> by Th5;
  then
A5: len Ant(f2) in dom Ant(f2) by Th10;
  (Ant(f)^<*'not' Suc(f)*>).(len Ant(f) + 1) = 'not' Suc(f) by FINSEQ_1:42;
  then (Ant(f2)).(len Ant(f) + 1) = 'not' Suc(f) by Th5;
  then (Ant(f2)).(len Ant(f) + 1) = Suc(f2) by Th5;
  then (Ant(f2)).(len Ant(f2)) = Suc(f2) by A4,FINSEQ_2:16;
  then
A6: |- f2 by A5,Lm2,Th33;
  set f4 = Ant(f1)^<*p*>;
  f2 = Ant(f1)^<*'not' Suc(f)*> by Th5;
  then
A7: f2 = Ant(f1)^<*'not' Suc(f1)*> by Th5;
  Ant(f1) = Ant(f)^<*'not' Suc(f)*> & Suc(f1) = Suc(f) by Th5;
  then |- f1 by A2,Th8,Th36;
  then
A8: |- Ant(f1)^<*p*> by A6,A7,Th44;
  set f3 = f^<*p*>;
  1+1 <= len f + 1 by A1,XREAL_1:6;
  then 1+1 <= len f + len <*p*> by FINSEQ_1:39;
  then 1+1 <= len f3 by FINSEQ_1:22;
  then
A9: 1 < len f3 by NAT_1:13;
A10: Ant(f1) = Ant(f)^<*'not' Suc(f)*> by Th5;
  then Suc(Ant(f1)) = 'not' Suc(f) by Th5;
  then Suc(Ant(f1)) = 'not' Suc(Ant(f3)) by Th5;
  then
A11: Suc(Ant(f4)) = 'not' Suc(Ant(f3)) by Th5;
  1 <= len Ant(f1) by A10,Th10;
  then 1+1 <= len Ant(f1) + 1 by XREAL_1:6;
  then 1+1 <= len Ant(f1) + len <*p*> by FINSEQ_1:39;
  then 1+1 <= len f4 by FINSEQ_1:22;
  then
A12: 1 < len f4 by NAT_1:13;
  Ant(f4) = Ant(f1) by Th5;
  then Ant(Ant(f4)) = Ant(f) by A10,Th5;
  then
A13: Ant(Ant(f4)) = Ant(Ant(f3)) by Th5;
  Suc(f4) = p by Th5;
  then |- Ant(Ant(f3))^<*Suc(f3)*> by A3,A8,A9,A12,A13,A11,Th5,Th37;
  then |- Ant(f)^<*Suc(f3)*> by Th5;
  hence thesis by Th5;
end;
