reserve A for QC-alphabet;
reserve p, q, r, s, p1, q1 for Element of CQC-WFF(A),
  X, Y, Z, X1, X2 for Subset of CQC-WFF(A),
  h for QC-formula of A,
  x, y for bound_QC-variable of A,
  n for Element of NAT;

theorem Th38:
  p is_an_universal_closure_of q implies p |-| q
proof
  assume p is_an_universal_closure_of q;
  then consider n being Element of NAT such that
A1: 1 <= n and
A2: ex L being FinSequence st len L = n & L.1 = q & L.n = p &
 for k being Nat st 1 <= k & k < n ex x, r st r = L.k & L.(k+1) = All(x,r);
  consider L being FinSequence such that
  len L = n and
A3: L.1 = q and
A4: L.n = p and
A5: for k being Nat st 1 <= k & k < n holds ex x, r st r = L.
  k & L.(k+1) = All(x,r) by A2;
  for k being Nat st 1 <= k & k <= n ex r st r = L.k & q |-| r
  proof
    defpred P[Nat] means 1 <= $1 & $1 <= n implies ex r st r = L.$1 & q |-| r;
A6: for k being Nat st P[k] holds P[k+1]
    proof
      let k be Nat such that
A7:   P[k];
      now
        assume that
        1 <= k+1 and
A8:     k+1 <= n;
        per cases by A8,NAT_1:13,14;
        case
A9:       k = 0;
          take p=q;
          thus ex r st r = L.(k+1) & q |-| r by A3,A9;
        end;
        case
A10:      1 <= k & k < n;
          then consider x, r such that
A11:      r = L.k and
A12:      L.(k+1) = All(x,r) by A5;
          consider s such that
A13:      s = All(x,r);
          s |-| r by A13,Th36;
          hence ex s st s = L.(k+1) & q |-| s by A7,A10,A11,A12,A13,Th28;
        end;
      end;
      hence thesis;
    end;
A14: P[0];
    thus for k being Nat holds P[k] from NAT_1:sch 2(A14,A6 );
  end;
  then ex r st r = L.n & q |-| r by A1;
  hence thesis by A4;
end;
