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
  (ex y st |- f^<*p.(x,y)*>) implies |- f^<*Ex(x,p)*>
proof
  given y such that
A1: |- f^<*p.(x,y)*>;
  set f1 = f^<*All(x,'not' p)*>^<*All(x,'not' p)*>;
A2: Ant(f1) = f^<*All(x,'not' p)*> by Th5;
  len f+1 = len f + len <*All(x,'not' p)*> by FINSEQ_1:39;
  then len f+1 = len Ant(f1) by A2,FINSEQ_1:22;
  then
A3: len f+1 in dom Ant(f1) by A2,Th10;
A4: Suc(f1) = All(x,'not' p) by Th5;
  (Ant(f1)).(len f+1) = All(x,'not' p) by A2,FINSEQ_1:42;
  then Suc(f1) is_tail_of Ant(f1) by A4,A3,Lm2;
  then |- f^<*All(x,'not' p)*>^<*('not' p).(x,y)*> by A2,A4,Th33,Th42;
  then |- f^<*All(x,'not' p)*>^<*'not' (p.(x,y))*> by Th56;
  then |- f^<*p.(x,y)*>^<*'not' All(x,'not' p)*> by Th49;
  then
A5: |- f^<*p.(x,y)*>^<*Ex(x,p)*> by QC_LANG2:def 5;
  1 <= len (f^<*p.(x,y)*>) by Th10;
  then |- Ant(f^<*p.(x,y)*>)^<*Ex(x,p)*> by A1,A5,Th45;
  hence thesis by Th5;
end;
