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 Th31:
  for X holds {p: ex f st f is_a_proof_wrt X & Effect(f) = p} = Cn(X)
proof
  let X;
  set PX = {p: ex f st f is_a_proof_wrt X & Effect(f) = p};
A1: PX c= Cn(X) by Lm12;
  reconsider PX as Subset of CQC-WFF(Al) by Lm2;
 X c= PX by Th29;
then
 Cn(X) c= {p: ex f st f is_a_proof_wrt X & Effect(f) = p} by Th12,Th30;
  hence thesis by A1;
end;
