reserve Al for QC-alphabet;
reserve i,j,n,k,l for Nat;
reserve a for set;
reserve T,S,X,Y for Subset of CQC-WFF(Al);
reserve p,q,r,t,F,H,G for Element of CQC-WFF(Al);
reserve s for QC-formula of Al;
reserve x,y for bound_QC-variable of Al;
reserve f,g for FinSequence of [:CQC-WFF(Al),Proof_Step_Kinds:];

theorem Th29:
  X c= {F: ex f st f is_a_proof_wrt X & Effect(f) = F}
proof
  let a be object;
  assume
A1: a in X;
  then reconsider p=a as Element of CQC-WFF(Al);
  reconsider pp=[p,0] as Element of [:CQC-WFF(Al),Proof_Step_Kinds:] by Th17,
ZFMISC_1:87;
  set f=<*pp*>;
A2: len f = 1 by FINSEQ_1:40;
(f.len f)`1 = p by A2;
then A4: Effect(f) = p by Def6;
 1 <= n & n <= len f implies f,n is_a_correct_step_wrt X
  proof
    assume 1 <= n & n <= len f;
then A5: n=1 by A2,XXREAL_0:1;
A6: (f.1)`2 = 0;
 (f.n)`1 in X by A1,A5;
    hence thesis by A5,A6,Def4;
  end;
then  f is_a_proof_wrt X;
  hence thesis by A4;
end;
